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Advance Computer Architecture

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Advanced Computer Architecture-CS501
________________________________________________________
Advanced Computer Architecture
Lecture No. 35
Reading Material
Vincent P. Heuring & Harry F. Jordan
Chapter 6
Computer Systems Design and Architecture
6.3, 6.4
Summary
Overflow
Different Implementations of the adder
Unsigned and Signed Multiplication
Integer and Fraction Division
Branch Architecture
Overflow
When two m-bit numbers are added and the result exceeds the capacity of an m-bit
destination, this situation is called an overflow. The following example describes this
condition:
Example 1
Overflow in fixed point addition:
In these three cases, the fifth position is not allowed so this results in an overflow.
Different Implementations of the Adder
For a binary adder, the sum bit is obtained by following equation:
__ _ _ __
sj = xjyjcj+xjyjcj+xjyjcj+xjyjcj
and the equation for carry bit is
cj+1=xjyj+xjcj+yjcj
where x and y are the input bits.
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The sum can be computed by the two methods:
Ripple Carry Adder
Carry Look ahead Adder
Ripple Carry Adder
In this adder circuit, we feed carry out from the previous stage to the next stage and so
on. For 64 bit addition, 126 logic levels are required between the input and output bits.
The logic levels can be reduced by using a higher base (Base 16). This is a relatively slow
process.
Complement Adder/Subtractor
We can perform subtraction using an unsigned adder by
 Complement the second input
 Supply overflow detection hardware
2's Complement Adder/Subtractor
A combined adder/subtractor can be built using a mux to select the second adder input. In
this case, the mux also determines the carry-in to the adder. The equation for mux output
is :
_  _
qj =y j r + yj r
Carry Look ahead Adder
The basic idea in carry look ahead is to speed up the ripple carry by determining whether
the carry is generated at the j position after addition, regardless of the carry-in at that
stage or the carry is propagated from input to output in the digit.
This results in faster addition and lesser propagation delay of the carry bits. It divides the
carry into two logical variables Gj (generate) and Pj (propagate). These variables are
defined as:
Gj = xjyj
P  j = x  j +y j
Hence the carry out will be
c  j +1= G  j +P j c j
Here the G and P each require one gate, and the sum bit needs two more gates in the full
adder. This results in a less complexity i.e. log(m) which is much less as compare to
ripple carry adder where complexity is m (m is the number of bits of a digit to be added).
Ripple carry and look ahead schemes are can be mixed by producing a carry-out at the
left end of each look ahead module and using ripple carry to connect modules at any level
of the look ahead tree.
Unsigned Multiplication
The general schema for unsigned multiplication in base b is shown in Figure 6.5 of the
text book.
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Parallel Array Multiplier
Figure 6.6 of the text book shows the structure of a fully parallel array multiplier for base
b integers. All signal lines carry base b digits and each computational block consists of a
full adder with an AND gate to form the product xiyj. In case of binary, m2 full adders are
required and the signals will have to pass through almost 4m gates.
Series parallel Multiplier
A combination of parallel and sequential hardware is used to build a multiplier. This
results in a good speed of operation and also saves the hardware.
Signed Multiplication
The sign of a product is easily computed from the sign of the multiplier and the
multiplicand. The product will be positive if both have same sign and negative if both
have different sign. Also, when two unsigned digits having m and n bits respectively are
multiplied, this results in a (m+n) ­bit product, and (m+n+1)-bit product in case of sign
digits. There are three methods for the multiplication of sign digits:
1. 2's complement multiplier
2. Booth recoding
3. Bit-Pair recoding
2's complement Multiplication
If numbers are represented in 2's complement form then the following three
modifications are required:
1. Provision for sign extension
2. Overflow prevention
3. Subtraction as well as addition of the partial product
Booth Recoding
The Booth Algorithm makes multiplication simple to implement at hardware level and
speed up the procedure. This procedure is as follows:
1. Start with LSB and for each 0 of the original number, place a 0 in the recorded
number until a 1 in indicated.
2. Place a 1 for 1in the recorded table and skip any succeeding 1's until a 0 is
encountered.
3. Place a 0 with 1 and repeat the procedure.
Example 2
Recode the integer 485 according to Booth procedure.
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Solution
Original number:
00111100101=256+128+64+32+4+1=485
Recoded Number:
_  _ _
01000101111=+512-32+8-4+2-1=485
Bit-Pair Recoding
Booth recoding may increase the number of additions due to the number of isolated 1s.
To avoid this, bit-pair recoding is used. In bit-pair recoding, bits are encoded in pairs so
there are only n/2 additions instead of n.
Division
There are two types of division:
Integer division
Fraction division
Integer division
The following steps are used for integer division:
1. Clear upper half of dividend register and put dividend in lower half. Initialize
quotient counter bit to 0
2. Shift dividend register left 1 bit
3. If difference is +ve, put it into upper half of dividend and shift 1 into quotient. If ­
ve, shift 0 into quotient
4. If quotient bits<m, goto step 2
5. m-bit quotient is in quotient register and m-bit remainder is in upper half of
dividend register
Example 3
Divide 4710 by 510.
Solution
D=000000 101111,
d=000101
D
000001 011110
d
000101
Diff(-)
q
0
D
000010 111100
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d
000101
Diff(-)
q
00
D
000101 111000
d
000101
Diff(+)
q
001
D
000001 110000
d
000101
Diff(-)
q
0010
D
000011 100000
d
000101
Diff(-)
q
00100
D
000111 000000
d
000101
Diff(+)000010
q
001001
Hence remainder = (000010)2 = 210
Quotient = (001001)2 = 910
Fraction Division
The following steps are used for fractional division:
1. Clear lower half of dividend register and put dividend in upper half. Initialize
quotient counter bit to 0
2. If difference is +ve, report overflow
3. Shift dividend register left 1 bit
4. If difference is +ve, put it into upper half of dividend and shift 1 into quotient. If
negative, shift 0 into quotient
5. If quotient bits<m, go to step 3
6. m-bit quotient has decimal at the left end and remainder is in upper half of
dividend register
Branch Architecture
The next important function perform by the ALU is branch. Branch architecture of a
machine is based on
1. condition codes
2. conditional branches
Condition Codes
Condition Codes are computed by the ALU and stored in processor status register. The
`comparison' and `branching' are treated as two separate operations. This approach is not
used in the SRC. Table 6.6 of the text book shows the condition codes after subtraction,
for signed and unsigned x and y. Also see the SRC Approach from text book.
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Usually implementation with flags is easier however it requires status registers. In case of
branch instructions, decision is based on the branch itself.
Note: For more information on this topic, please see chapter 6 of the text book.
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Table of Contents:
  1. Computer Architecture, Organization and Design
  2. Foundations of Computer Architecture, RISC and CISC
  3. Measures of Performance SRC Features and Instruction Formats
  4. ISA, Instruction Formats, Coding and Hand Assembly
  5. Reverse Assembly, SRC in the form of RTL
  6. RTL to Describe the SRC, Register Transfer using Digital Logic Circuits
  7. Thinking Process for ISA Design
  8. Introduction to the ISA of the FALCON-A and Examples
  9. Behavioral Register Transfer Language for FALCON-A, The EAGLE
  10. The FALCON-E, Instruction Set Architecture Comparison
  11. CISC microprocessor:The Motorola MC68000, RISC Architecture:The SPARC
  12. Design Process, Uni-Bus implementation for the SRC, Structural RTL for the SRC instructions
  13. Structural RTL Description of the SRC and FALCON-A
  14. External FALCON-A CPU Interface
  15. Logic Design for the Uni-bus SRC, Control Signals Generation in SRC
  16. Control Unit, 2-Bus Implementation of the SRC Data Path
  17. 3-bus implementation for the SRC, Machine Exceptions, Reset
  18. SRC Exception Processing Mechanism, Pipelining, Pipeline Design
  19. Adapting SRC instructions for Pipelined, Control Signals
  20. SRC, RTL, Data Dependence Distance, Forwarding, Compiler Solution to Hazards
  21. Data Forwarding Hardware, Superscalar, VLIW Architecture
  22. Microprogramming, General Microcoded Controller, Horizontal and Vertical Schemes
  23. I/O Subsystems, Components, Memory Mapped vs Isolated, Serial and Parallel Transfers
  24. Designing Parallel Input Output Ports, SAD, NUXI, Address Decoder , Delay Interval
  25. Designing a Parallel Input Port, Memory Mapped Input Output Ports, wrap around, Data Bus Multiplexing
  26. Programmed Input Output for FALCON-A and SRC
  27. Programmed Input Output Driver for SRC, Input Output
  28. Comparison of Interrupt driven Input Output and Polling
  29. Preparing source files for FALSIM, FALCON-A assembly language techniques
  30. Nested Interrupts, Interrupt Mask, DMA
  31. Direct Memory Access - DMA
  32. Semiconductor Memory vs Hard Disk, Mechanical Delays and Flash Memory
  33. Hard Drive Technologies
  34. Arithmetic Logic Shift Unit - ALSU, Radix Conversion, Fixed Point Numbers
  35. Overflow, Implementations of the adder, Unsigned and Signed Multiplication
  36. NxN Crossbar Design for Barrel Rotator, IEEE Floating-Point, Addition, Subtraction, Multiplication, Division
  37. CPU to Memory Interface, Static RAM, One two Dimensional Memory Cells, Matrix and Tree Decoders
  38. Memory Modules, Read Only Memory, ROM, Cache
  39. Cache Organization and Functions, Cache Controller Logic, Cache Strategies
  40. Virtual Memory Organization
  41. DRAM, Pipelining, Pre-charging and Parallelism, Hit Rate and Miss Rate, Access Time, Cache
  42. Performance of I/O Subsystems, Server Utilization, Asynchronous I/O and operating system
  43. Difference between distributed computing and computer networks
  44. Physical Media, Shared Medium, Switched Medium, Network Topologies, Seven-layer OSI Model