The assembly language that has been studied so far is not a form that the processor can understand.
Therefore, we need to encode instructions and data in well-formed binary.

Data representation

numbers

• Numbers are kept in computer hardware as a series of 1 and 0
• They are considered base 2 numbers (binary numbers)
• Binary numbers are stored in words
• In MIPS, the words are 34 bits (4 bytes) long + MIPS is big endian

unsigned numbers

• By using n bits, we can represent unsigned numbers from 0 to $2^n-1$

signed numbers

Signed Magnitude

• first bit determines mathematical symbols
  • 0 is plus (+)
  • 1 is minus (-)
• others are bit size
• but, 000 = 100 = 0

One's complement

• first bit determines mathematical symbols
  • 0 is plus (+)
  • 1 is minus (-)
• if first bit is 0
  • read it as it is
• if first bit is 1
  • flip 1 to 0, 0 to 1
• ex) 100 = -3
• but, 000 = 111 = 0

Two's complement

• first bit determines mathematical symbols
  • 0 is plus (+)
  • 1 is minus (-)
• if first bit is 0
  • read it as it is
• if first bit is 1
  • flip 1 to 0, 0 to 1
  • plus 1
• ex) 100 = -4
• the number of zero is 1

Answer

two's complement

• we can get the computation result by just doing given arithmetic operations
  • 000 - 001 = 111 ( 0 - 1 = -1)
  • 010 + 111 = 001 (2 + (-1) = 1)
• the number of zero is only 1.
• by using n bits, we can represent signed numbers from $-2^{n-1}$ to $2^{n-1}-1$

Signed extension

Sometimes, we need to represent n-bit numbers by using more than n bits

• 16-bit immediate should be converted to 32 bits for arithmetic
• Instructions lb/lh loads byte/halfword from memory space and store it into 32-bit registers
• Replicate the sign bit to the left

Instruction representation

Like data, instructions are also encoded/represented in binary
We call the encoded instructions as machine instructions

For representing instructions, ISA defines instruction format
Issue: to represent all kinds of instructions, we might need many instrucion formants

Design principle 4

Good design demands good compromise

Based on this, MIPS keeps formats as similar as possible (regularity)

R-format

For the instructions that use only Register operands

• op (opcode): basic operation of the instruction (what the instruction does)
• rs: the first source register operand
• rt: the second source register operand
• rd: the destination register operand
• shamt: shift amount (used for shift operations)
• funct: function code (the specific variant of the operation)

Q. why are the rs, rt, rd 5 bits?
A. registers are 32, which means 5 bits are enough to express each register

I-format

For the instructions that use Immediate operands

• op (opcode): basic operation of the instruction (what the instruction does)
• rs: the first source register operand
• rt: the second source register operand
• Constant or address

Summary

Key underlying design principles

Design Principle 1

Simplicity favors regularity

All MIPS arithmetic instructions include a single operation & three operands

• Lower clock period or CPI

Design Principle 2

Smaller is faster

Operands of MIPS arithmetic instructions must be chosen in a small number of registers.
MIPS keeps more complex data in memory and supports data transfer between memory and registers.

• Lower clock period or CPI

Design Principle 3

Make the common case fast

Support 16-bit immediate operands for handling small constants + $zero

• Lower Instruction count

Design Principle 4

Good design demands goog compromise

Keep all instructions the same length + keep instruction formats similar as possible.
Data (numbers) are also represented in binary based on two's complement rules.

• Lower clock period or CPI