1. Arithmetic operations in binary
Addition / subtraction
01011 ?
“Method” exatly the same as decimal

2. Arithmetic operations in binary
Addition
X = xn … xi … x0
+ Y = yn … yi … y0
__________________ di
di = (xi + yi ) mod r + carry-in

3. Addition For ( i = 0…n ) do di = (xi + yi + carry-in) mod r
carry-out = (xi + yi + carry-in) div r
End for

4. 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

5. Arithmetic operations in binary
Subtraction
X = xn … xi … x0
 Y = yn … yi … y0
___________________________________________________ di
For ( i = 0…n ) do
di = ( xi  yi  bi-1 ) mod r
borrow-out = ( xi  yi  bi-1 ) div r
End for

6. Subtraction

7. Arithmetic operations in binary
Multiplication
For ( i = 0…n ) do
di = ( xi  yi + ci-1 ) mod r
ci = ( xi  yi + ci-1 ) div r
End for
Is this correct?

8. Negative numbers (4 traditions):
Signed magnitude
Radix complement
Diminished radix complement
Excess-b (biased)
n = number of symbols (digits, bits, etc.)
r = radix
radix complement of x is
e.g. n = 4, r = > = 2784 (10s complement)
n = 4, r = > = 1011 (2s complement)

9. diminished radix complement is
e.g. n = 4, r = > = (9s complement)
n = 4, r = > = (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
9s complement: > > 0000

10. Arithmetic with complements n = 4 r = 2
Excess-8

11. 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 – = (67454)10

12. Two’s complement arithmetic
( an-1, an-2, … a0 ) in 2-s complement is
Example: (n = 4)
0101  023 + 122 + 02 + 11 = = 5
1011  123 + 022 + 12 + 11 =  = 5
Addition/subtraction (n = 5)
+10  01010
+3  00011
01101  13

13. Two’s complement arithmetic
Addition/subtraction (n = 5)
+10 
+7 
Overflow  15
+15 
13 
Discard  2
When can overflow happen? Only when both operands have the same sign and the sign bit of the result is different.

14. 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)

15. How to detect discontinuity?5
+ 6 11
0101
1011
no overflow (in 4-bit register)
but, carry into most significant bit is 1 while carry out is 0 -5
-11
1011
10101
overflow
but, carry into most significant bit is 0 while carry out is 1

16. Same circuitry. signed numbers. add
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 detect with overflow for addition lack of overflow for subtraction
Intel uses CF (carry flag) to detect out-of-range result

17. Codes 4-bit codes for digits 0 through 9 (BCD : 8421 weighted)
2421 and Excess-3 are self-complementing
(complement bits to get 9’s complement representation)
Biquinary has two ranges 01… and 10…
one-bit error produces invalid codeword

19. Number representation inside computer
Integers – represented in 2’s complement
Single precision – 32 bits
Largest positive integer is =
2,147,483,647
Smallest negative integer is =
2,147,483,648

20. Number representation inside computer
Floating point
Scientific notation
= 10-2
normalized number
The digit after the decimal point is  0
Normalized notation maximizes the use of significant digits.

21. Floating point numbers
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

22. Floating point formats
IEEE format: (-1)S(1.m)2E-127
DEC format: (-1)S(0.m)2E-128

23. Floating point formats
Different manufacturers used different representations

24. Mechanical encoder with sequential
sector numbering
At boundaries can get a code different
than either adjacent sector

25. Gray code:

26. Gray code algorithm: input: (binary) output: (Gray code) e.g.
Alternative algorithm (n bits)
If n = 1 use 0->0 and 1->1
If n > first half is Gray code on (n - 1) bits preceded by 0
second half is reversed Gray code on (n - 1) bits preceded by 1

27. 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

28. Codes

29. 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

30. Ascii, ebcdic codes
State codes, e.g.

31. Consequences of binary versus one-hot coding:

32. 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

33. Gray code is path that visits each
vertex exactly once

34. 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
Problem
Solution: parity

35. Error-correcting codes minimum distance between code words > 1
Example: correct one bit errors or detect two-bit errors
Two-bit error from

36. Write minimum distance as 2c + d + 1 bits ==>
corrects c-bit errors and
detects (c + d)-bit errors
Example: min distance = 4 = 2(1)
But also, 4 = 2(0)

37. 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

38. Create group for each check bit
Hamming codes:
number bit positions 1, 2, 3, ... n from right to left
bit position is a power of 2 => check bit
else => 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)

39. Code information packets to maintain even parity in groups
e.g. (n = 7)
packet is 1011 => positions 7, 6, 5, 3
1 0 1 x 1 x x
Consult group memberships to compute check bits
check information
1 3, 5, 7 => bit 1 is 1
2 3, 6, 7 => bit 2 is 0
4 5, 6, 7 => bit 4 is 0
Code word is

40. Note:
Single bit error corrupts one or more parity groups => minimum distance > 1
Two-bit error in locations x, y corrupts at least one parity group =>
minimum distance > 2
Three-bit error (i.e. 1, 4, 5) goes undetected => minimum distance = 3
3 = 2(1) = 2(0) => can correct 1-bit errors or detect errors
of size 1 or 2.

41. Pattern generalizes to longer bit strings:
Group 8
Group 4
Group 2
Group 1
Pattern generalizes to longer bit strings:
Single bit error corrupts one or more parity groups => minimum distance > 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 => min distance > 2
Three-bit error (two msb and lsb) goes undetected => minimum distance = 3
Conclude: process always produces a Hamming code with min distance = 3

42. Traditional to permute check bits to far right
Used in memory protection schemes

43. Add traditional parity bit (for entire word) to obtain
code with minimum distance = 4

44. Number of check bits grows slowly with information bits
For minimum distance = 4:
Number of check bits grows slowly with information bits

45. Two-dimensional codes (product codes)
min distance is product of
row and column distances
for simple parity on each (below)
min distance is 4

46. 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

47. 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

48. Serial data transmission
(simple) transmit clock and sync with data (3 lines)
(complex) recover clock and/or sync from data line

49. Serial line codes:
NRZ: clock recovery except during long sequences of zeros or ones
NRZ1: nonreturn to zero, invert on one zero => no change, one => 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 => 0 to 1 transition, one => 1 to 0 transition at interval center