Class

1
Class 3 Addressing Data Representation

• Addressing
• A. Modes
• B. Examples
• II. Binary Number Operations
• A. Numbers with radix point
• B. Floating Point Notation (IEEE 754 Standard)
• C. Floating Point Arithmetic Operations
• D. Multiplication Division
• III. Representing Pictures and Audio
• A. Digitizing Images -gt Binary
• B. Digitizing Audio - gt Binary
• IV. Data Structures
• A. Stacks
• B. Reverse Polish Notation (RPN)
• C. Big Endian Little Endian

2
Addressing Modes

• Immediate
• Direct
• Indirect
• Register
• Register Indirect
• Displacement (Indexed)

3
Immediate Addressing

• Operand is part of instruction
• e.g. ADD 5
• Add 5 to contents of accumulator
• 5 is operand
• No memory reference to fetch data
• Fast
• Limited range

Instruction
Operand
Opcode
4
Direct Addressing

• Common in early generations of computers
• Address field contains address of operand
• Effective address (EA) address field (A)
• e.g. ADD A
• Add contents of cell A to accumulator
• Look in memory at address A for operand
• Single memory reference to access data
• No additional calculations to work out effective
  address
• Limited address space

5
Direct Addressing Diagram
Instruction
Address A
Opcode
Memory
Operand
6
Indirect Addressing

• Memory cell pointed to by address field contains
  the address of (pointer to) the operand
• EA (A)
• Look in A, find address (A) and look there for
  operand
• e.g. ADD (A)
• Add contents of cell pointed to by contents of A
  to accumulator
• Large address space 2n where n word length
• Multiple memory accesses to find operand
• Hence slower

7
Indirect Addressing Diagram
Instruction
Address A
Opcode
Memory
Pointer to operand
Operand
8
Register Addressing

• Operand is held in register named in address
  field
• EA R
• Limited number of registers
• Very small address field needed
• Shorter instructions
• Faster instruction fetch
• No memory access very fast execution
• Very limited address space
• Multiple registers helps performance
• Requires good assembly programming or compiler
  writing
• N.B. C programming
• register int a
• c.f. Direct addressing

9
Register Addressing Diagram
Instruction
Register Address R
Opcode
Registers
Operand
10
Register Indirect Addressing

• C.f. indirect addressing
• EA (R)
• Operand is in memory cell pointed to by contents
  of register R
• Large address space (2n)
• One fewer memory access than indirect addressing

11
Register Indirect Addressing Diagram
Instruction
Register Address R
Opcode
Memory
Registers
Operand
Pointer to Operand
12
Displacement Addressing

• EA A (R)
• Address fields hold two values
• A base value
• R register that holds displacement
• or vice versa
• Flavors
• PC Relative addressing
• Base-register addressing
• Indexing

13
Displacement Addressing Diagram
Instruction
Address A
Register R
Opcode
Memory
Registers
Pointer to Operand
Operand

14
PC-Relative Addressing

• A version of displacement addressing
• R Program counter, PC
• EA A (PC)
• i.e. get operand from A cells from current
  location pointed to by PC
• Often used with branch instruction

15
Base-Register Addressing

• A holds displacement
• R holds pointer to base address
• R may be explicit or implicit
• e.g. segment registers in 80x86

16
Indexed Addressing

• A base
• R displacement
• EA A R
• Good for accessing arrays
• EA A R
• R (increment next element in array)

17
Pentium Addressing Modes

• Virtual or effective address is offset into
  segment
• Starting address plus offset gives linear address
• This goes through page translation if paging
  enabled
• 12 addressing modes available
• Immediate
• Register operand
• Displacement
• Base
• Base with displacement
• Scaled index with displacement
• Base with index and displacement
• Base scaled index with displacement
• Relative

18
Addressing Example

• Q If mode indicates a source register it is R1
  which has a value of 200. There is also a base
  register that contains the value 100. Location
  499 contains the value 999, location 500 contains
  the value 1000 and so on. What is the effective
  address and operand for the following?

Answer
19
Binary Real Numbers

• A real number can contain both whole and
  fractional components
• 18.0 divided by 4.0 4.5
• 10010 / 100 100.1 (can be done by long
  division)
• We can represent non-integer numbers in binary by
  use of a radix point.
• Q 1 What is 96.375
• in binary?
• Q 2 What is 11001.1101
• in decimal?

20
Adding Numbers with Radix Points

• What is 101101.101 10100.1010?
• You have align the radix points in order to get
  the answer
• 101101.101 45.625
• 10100.1011 20.6875
• 1000010.0101 66.3125
• (ones carry from right of radix point to left
  of it)

21
Floating Point Notation

• Floating point notation is used to represent very
  small numbers and very large numbers
• The exponent attached to the base can be
  interpreted as the number and the direction of
  positional moves of the radix point
• Negative exponents indicate a movement to the
  left, and positive exponents indicate movement to
  the right

22
Floating Point Numbers

• Components
• Sign bit
• Exponent
• Assumed radix point
• Assumed (hidden) 1
• Mantissa

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The Sign Mantissa

• The Sign
• A 1 bit indicates a negative number, and a 0 bit
  indicates a positive number.
• The Mantissa
• Decimal 0.154 1/10 5/100 4/1000
• Binary 0.1011 1/2 0/4 1/8 1/16
• For the above binary number, we have a
  normalized binary number of 1.011 x 2-1
• The sign is positive (0), the exponent is -1 and
  the mantissa for an IEEE 754 floating point
  number is 011.
• Why is the mantissa NOT 1011?

24
The Exponent

• IEEE 754 Short (32-bit) real exponents are
  stored as 8-bit unsigned integers with a bias of
  127.

Actual Exponent Adjusted Exponent - 127
25
Normalized Mantissa

• What are the following floating point numbers in
  binary decimal?
• 10111111 00000000 00000000 00000000
• Negative, exp126-127 -1, mantissa 0, 1.0 x
  2-1 - (0.10)2 -(1/2)10
• 00111111 10000000 00000000 00000000
• Positive, exp127-127 0, mantissa 0, 1.0 x
  20 (1)2 (1)10
• 3F880000H 00111111 10001000 00000000 00000000
• Positive, exp0, mantissa 0001, 1.0001 x 20
  (1.0001)2 (1.0625)10

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Single Precision Double Precision

• All 1s in exponent infinity
• All 0s in exponent zero

27
Problems with Floating Point s

• Overflow occurs when the exponent is larger than
  the allocated space.
• Underflow occurs when a negative exponent is too
  large in absolute value to fit within the bits
  allocated to store it.
• Truncation occurs when the there are not enough
  digits in the mantissa to represent the number.
• Examples
• 1/3 in decimal 0.3333333...
• 1/10 cannot be represented accurately as a
  floating point number (many others also). Problem
  with !

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Floating Point Arithmetic

• Floating point addition / subtraction
• Floating point multiplication/ division
• Floating point arithmetic differs from integer
• arithmetic because exponents and fraction must
  be
• handled separately differently.

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Floating Point Addition/Subtraction

• The exponents of the operands must be made equal
  for addition and subtraction. The fractions are
  then added or subtracted as appropriate. The
  result is normalized.
• Example (.10123 .111 24)2
• Start by adjusting the smaller exponent to be
  equal to the larger exponent, and modify the
  fraction accordingly. Thus we have .101 23
  .0101 24.
• The resulting sum is (.0101 .1110) 24
  1.001124 .1001125
• 5 14 19
• Note, the hidden bit 1 is made explicit
  during calculations.

30
Floating Point Addition/Subtraction

• Procedures
• Check for zeros
• Adjust exponents and align significands
• Add or subtract significands
• Normalize result

31
Floating Point Addition/Subtraction Flowchart
32
Unsigned Multiplication

• Multiplication involves the generation of partial
  products.
• Each successive partial product is shifted one
  position to the left relative to the preceding
  partial product.
• The total result is produced by summing the
  partial products.
• Multiplication of two n-bit unsigned binary
  integers produces a 2n-bit result.

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Flowchart for Unsigned Binary Multiplication
34
Unsigned Division

• Example 147/11 13 remainder 4
• Examine the dividend from left to right, until
  the set of bits is greater than or equal to the
  divisor.
• Place 1 in the quotient and subtract the
  divisor from the partial dividend. The result is
  partial remainder.
• Additional bits from the dividend are appended
  to the partial remainder until the partial
  remainder is greater than or equal to the
  divisor.
• Repeat step 2, until all the bits of the
  dividend are exhausted.

35
Flowchart for Unsigned Binary Division
36
Multiplication Division of Signed Integers

• The general technique for multiplication and
  division of signed integers is to convert the
  operands into their positive forms, perform the
  division, and then convert the result into its
  true signed form as a final step.
• Booths algorithm is a way to multiply signed
  numbers.

37
Floating Point Multiplication/Division

• For floating point multiplication/division, the
  sign, exponent and fraction of the
• result are computed separately.
• Sign Like/unlike signs produce
  positive/negative results.
• Exponent Adding exponents for
  multiplication.
• Subtracting exponents for
  division.
• Fractions Multiplying or dividing as for
  fix point numbers.
• Normalizing the final result.
• Example (.110 25) / (.100 24)2
• The source operand signs are the same, so the
  result has a positive sign.
• Subtracting exponents for division 5 4 1.
• Dividing fractions 110/100 1.10.
• Putting it all together (.110 25) / (.100
  24) (1.10 21).

38
Representing Images
Each pixel color is represented by a certain
binary string. The number of pixels determines
the resolution.
39
Color Bit Depth per Pixel

• RGB One byte for each primary color

40
Example

• Q1 For a bit-map terminal with 1024 x 768 pixel
  display and 24-bit color, what is the data rate
  for an uncompressed video at 30 frames/sec?
• A 1024x768 786,432 pixels, with 24-bits/pixel
  we have 786,432 x 24 18,874,368 bits/frame
• With 30 frame/sec x 18,874,368 bits/frame
• 566,231,040 bits/sec approx. 566Mbps

41
Representing Audio Signals

• You have done this already!
• Remember Pulse Code Modulation (PCM)?
• Sampling (Nyquist rate)
• Quantization (what does this affect?)
• Encoding (what can we do here to improve
  reliability?)

42
Stacks

• Contiguous block of reserved space in main memory
• A data structure for variable numbers of items.
  Used because of not enough registers in CPU- so
  main memory is used instead.
• Stack is a LIFO (last in, first out) queue.
• - Stack expands as data is added
• - Stack contracts as data is removed

43
Stack Operation
44
Stack Organization

• Stack Limit End of block in main memory (usually
  lowest address in reserved space)
• Stack Pointer Address of top of stack
• Stack Base Address of bottom of stack (usually
  highest address in reserved space)
• Usually top two stack elements in registers (WHY?)

45
Stack Expression Evaluation
46
Reverse Polish Notation (postfix)

• Polish Notation was invented in the 1920's by
  Polish mathematician Jan Lukasiewicz, who showed
  that by writing operators in front of their
  operands, instead of between them, brackets were
  made unnecessary.
• Reverse Polish Notation puts the operators at the
  end (Dijkstra). It is used in stack operations.
• a b becomes ab
• a (b x c) becomes abcx
• (a b) x c becomes abcx

47
Byte Order

• What order do we read numbers that occupy more
  than one byte?
• e.g. (numbers in hex to make it easy to read)
• 12345678 can be stored in 4x8bit locations as
  follows
• Do we read top down or bottom up??

48
Byte Order Names

• The problem is called Endian
• The system on the left has the most significant
  byte in the lowest address for number 12345678
• This is called big-endian
• The system on the right has the least
  significant byte in the lowest address for number
  12345678
• This is called little-endian

Little used by Intel, VAX, Alpha
Big used by IBM, Motorola, Sun, RISC
49
Example of C Data Structure
50
Alternative View of Memory Map
51
Endian Example

• Big endian versus little endian

Name JIM SMITH Age 21 Dept 260
52
StandardWhat Standard?

• Pentium (80x86), VAX are little-endian
• IBM 370, Motorola 680x0 (Mac), and most RISC are
  big-endian
• Internet is big-endian
• Makes writing Internet programs on PC more
  awkward!
• WinSock provides htoi and itoh (Host to Internet
  Internet to Host) functions to convert