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Digital Circuits and Logic. Gates Combinatorial logic adder ALU Sequential logic flip-flop memory CPU fetch-decode-execute cycle. How to make a computer. Computers are electronic equipment contain many connected electronic components memory Central Processing Unit ( CPU )

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digital circuits and logic
Digital Circuits and Logic
  • Gates
  • Combinatorial logic
    • adder
    • ALU
  • Sequential logic
    • flip-flop
    • memory
  • CPU
    • fetch-decode-execute cycle

CMPUT 229

how to make a computer
How to make a computer
  • Computers are electronic equipment
    • contain many connected electronic components
      • memory
      • Central Processing Unit (CPU)
      • specialized devices (network, video, etc.)
    • components largely made of circuits containing wires and gates

CMPUT 229

logic level
Logic Level
  • A Boolean logical signal always takes one of two logic levels.
    • on / off
    • high (H) / low (L)
    • one (1) / zero (0)
    • true (T) / false (F)
    • positive / negative
  • Why two levels ?
    • Why not use, say, 4 levels?
      • For example, electrical voltage as
        • 0 v, 5v, 10v, and 20v

CMPUT 229

gates
Gates
  • Tiny electronic switches
    • made of transistors
    • implement Boolean logic
      • 0/1  low/high voltage
      • digital
    • for given inputs, produce known output
      • according to truth table
    • represented in circuit diagrams by symbols

CMPUT 229

gates5
Gates

NOT

In

Out

0

1

1

0

CMPUT 229

gates6
AND

In 1

In 2

Out

0

0

0

0

1

0

1

0

0

1

1

1

Gates

CMPUT 229

gates7
Gates

NAND

In 1

In 2

Out

0

0

1

0

1

1

1

0

1

1

1

0

CMPUT 229

gates8
In 1

In 2

Out

0

0

0

0

1

1

1

0

1

1

1

1

Gates

OR

CMPUT 229

gates9
XOR

In 1

In 2

Out

0

0

0

0

1

1

1

0

1

1

1

0

Gates

CMPUT 229

implementation of logic gates
Implementation of Logic Gates
  • The NAND gate, the basic logic gate
    • easier to manufacture
    • any other logic gate can be made from a combination of NAND gates

CMPUT 229

implementation of logic gates12
Implementation of Logic Gates
  • CMOS
    • The standard, 4000 series (4011)

CMPUT 229

implementation of logic gates13
Implementation of Logic Gates
  • TTL
    • The 4011's TTL counterpart is the 7400

CMPUT 229

logic circuits
Logic Circuits
  • Combinatorial Logic
    • Output depends solely on the input values
    • Example:
      • arithmetic logic unit (ALU)
  • Sequential logic
    • output depends on
      • the present input, and
      • the history of the input
    • Examle:
      • Flip-flop, Memory

CMPUT 229

combinatorial logic
Combinatorial logic
  • What about more complex truth tables?
  • Made by combining many gates together
    • wires connect inputs and outputs
    • each wire can carry only one bit, 0 or 1

CMPUT 229

combinatorial logic16
Combinatorial logic
  • Circuits made from collections of gates
    • outputs depend only on inputs
      • not on prior state
    • characterized by truth table
    • comparable to Boolean expression

CMPUT 229

combinatorial logic17
0

A

1

B

X

1

C

Combinatorial logic

boolean expression

X = (A & ~B) | (B & C)

truth table

logic circuit

0

0

0

0

1

1

1

1

1

all three of forms

are equivalent; each can be converted to the other two.

CMPUT 229

slide19
In 1

In 2

Out

0

0

0

0

1

1

1

0

1

1

1

1

CMPUT 229

combinatorial logic20
Combinatorial logic
  • Many parts of a computer are constructed of combinatorial logic
    • adder circuit
      • performs addition
    • arithmetic logic unit (ALU)
      • performs many kinds of arithmetic and bitwise logical operations
      • contains adder circuit
    • multiplexer and decoder
      • direct bit traffic between components

CMPUT 229

adder
Adder
  • A combinatorial circuit that adds binary values
    • according to this truth table

Carry is equivalent to AND

Sum is equivalent to XOR

CMPUT 229

adder22
Adder

Carry

In 1

In 2

Sum

This circuit can add two binary digits and is called a half-adder

CMPUT 229

adder23
1

5

3

+

4

3

8

Adder

Why “half-adder”. . . ?

1

9

1

. . . because to add multi-digit numbers each column requires two additions

CMPUT 229

full adder
half addersFull adder

in 1

This circuit is called a full adder and can add three binary digits

in 2

carry out

carry in

This OR gate combines the carries from the two half-adders

sum

CMPUT 229

ripple carry adder
Ripple-carry adder

A3

A2

A1

A0

B3

B2

B1

B0

out3

out2

out1

out0

CMPUT 229

ripple carry adder26
0

1

0

1

0

1

1

0

Ripple-carry adder

A3

A2

A1

A0

0101

+ 0110 = 1011

B3

B2

B1

B0

0

1

0

0

0

0

1

0

1

0

0

0

0

1

0

1

1

out3

out2

out1

out0

CMPUT 229

ripple carry adder27
Ripple-carry adder
  • So named because gate results (including carries) propagate (“ripple”) from LSB to MSB
    • right to left
    • corresponds to how humans add numbers with pen and paper
  • More sophisticated, faster, adder circuits exist that can create the higher order carries more quickly

CMPUT 229

adder28
Adder

A3-A0

B3-B0

Adders (and other arithmetic circuits) are usually drawn like this in block diagrams

inputs

+

output

collections of parallel, related wires like this are known as buses; they carry multi-bit values between components

out3-out0

CMPUT 229

arithmetic
Arithmetic
  • Computers need to do more than just addition
    • arithmetic: + – * / %
    • logic: & | ~ << >>
  • Need a circuit that can select operation to perform

CMPUT 229

multiplexer mux
Multiplexer (Mux)

B

A

A

B

S

S

C

C

CMPUT 229

arithmetic logic unit alu
Arithmetic Logic Unit (ALU)

A

B

more operations here

. . .

op 0

op 1

op 2

op 3

+

*

&

<<

Multiplexer: a combinatorial circuit which selects exactly one input

0

1

2

3

..

MUX

op

op selects operation:

0 = add, 1 = multiply, ...

out

CMPUT 229

arithmetic logic unit alu32
Arithmetic Logic Unit (ALU)

A = 15

B = 2

for example: compute 15 << 2

more operations here

. . .

op 0

op 1

op 2

op 3

+

*

&

<<

other results also computed but ignored by multiplexer

0

1

2

3

..

MUX

op = 3

out = 60

CMPUT 229

slide33
How to design a logic circuit for a given truth table?
    • For each row in the truth table whose output value is 1, construct a logic expression of the conjunction of all input column values
    • Form the disjunction of all conjunctive formulas obtained in step 1
    • Simplify the disjunction obtained from Step 2
    • Construct the logic circuit from the simplified logical expression in Step 3

CMPUT 229

memory
Memory
  • Computers need memory for storage
  • Different kinds of memory distinguished by speed, size, cost and proximity to CPU
    • main memory
      • slowish, huge, cheap, far from CPU
      • typical size 109 bits
    • cache
      • fast, medium-sized, expensive, near to CPU
      • typical size 106 bits
    • registers
      • extremely fast, tiny, very expensive, located on CPU
      • in MIPS (32 GPRs)×32 bits = 1024 bits

CMPUT 229

memory35
Memory
  • The smallest piece of memory is a single binary digit (bit)
    • can hold 0 or 1 only
  • A one-bit memory is called a flip-flop or a data latch
    • because its value can flip and flop between 0 and 1
    • because it can latch onto a data value and store it

CMPUT 229

flip flop
Flip-flop
  • Flip-flop needs two operation modes
    • write: store (memorize) a value
    • read: load (recall) a previously stored value
  • Also need
    • data in
      • for telling flip-flop what value to store (0 or 1)
      • used only when writing
    • data out
      • for finding out what value flip-flop currently contains (0 or 1)
      • used only when reading

CMPUT 229

flip flop37
Flip-flop
  • Flip-flop can be implemented with gates
  • Not combinatorial logic
    • because current output may depend on previous state
  • Example of sequential logic
    • current output depends on inputs and prior output

CMPUT 229

flip flop38
Flip-flop

NOR gate: OR gate followed by NOT gate

data in

data out

read/write

read/write control: 0 = read, 1 = write

CMPUT 229

flip flop writing
Flip-flop: writing

Try changing data in to 0 and watch data out

data in = 1

1

1

0

1

1

0

1

0

1

0

data out = 1

1

when read/write = 1, data out = data in

read/write = 1 (write)

CMPUT 229

flip flop reading
when read/write = 0, no signals in box can change, data out holds value regardless of data inFlip-flop: reading

data in = ?

?

0

0

0

1

0

?

?

1

0

data out = 1

0

read/write = 0 (read)

CMPUT 229

flip flop41
Flip-flop

D = data in

Q = data out

Flip-flops are often drawn like this in block diagrams

D

Q

CK

CK is read/write (“clock” because this input is often connected to computer’s processor clock)

CMPUT 229

flip flop using nand gates43
Flip-flop using NAND Gates

Data In

Data Out

read/write

CMPUT 229

memory44
Memory
  • Memory can store many bits independently
    • many flip-flops
  • Need to identify which bit (flip-flop) to read or write
  • Give each flip-flop a unique number (address)

CMPUT 229

memory45
Memory

Decoder: feeds input to selected output, 0 to all others

data in

D

Q

address 0

CK

D

Q

data out

0

0

address 1

rd/wr

1

1

CK

DEC

MUX

2

2

3

3

D

Q

...

...

address 2

CK

D

Q

address

address 3

CK

. . . millions more flip-flops . . .

CMPUT 229

memory writing
Memory: writing

Writing value 1 to flip-flop at address 2

data in

?

D

Q

1

address 0

0

CK

?

D

Q

data out

0

0

address 1

rd/wr

0

1

1

CK

DEC

MUX

2

2

1

1

3

3

D

Q

1

...

...

address 2

1

CK

?

D

Q

address

address 3

0

CK

2

. . . millions more flip-flops . . .

CMPUT 229

memory reading
Memory: reading

Reading value from flip-flop at address 2

data in

D

Q

address 0

0

CK

D

Q

data out

0

0

address 1

rd/wr

0

1

1

CK

DEC

MUX

2

2

0

1

1

3

3

D

Q

1

...

...

address 2

0

CK

D

Q

address

address 3

0

CK

2

. . . millions more flip-flops . . .

CMPUT 229

memory48
Memory
  • Memory usually operates in terms of bytes (8 bits), not single bits
  • Repeat memory circuit eight times
    • connect each memory circuit to one of the eight lanes of the data bus
    • reads and writes occur in parallel for each bit in byte

CMPUT 229

central processing unit cpu
Central Processing Unit (CPU)
  • Coordinates all computer’s components according to program being run
  • Contains
    • registers
    • ALU
    • program counter (PC)
      • address of current instruction
    • instruction register (IR)
      • copy of current instruction
    • control logic
  • Runs programs using fetch-(decode)-execute cycle

CMPUT 229

slide50
CPU

IR

Registers

out

Memory

in

Control

logic

rd/wr

addr

ALU

PC

CMPUT 229

fetch execute cycle fetch
Fetch-execute cycle: fetch

IR

Registers

out

Stage 1 of fetch-execute cycle: instruction at address pointed to by PC is fetched into IR

Memory

in

Control

logic

rd/wr

addr

ALU

PC

CMPUT 229

fetch execute cycle decode
Fetch-execute cycle: decode

IR

Registers

out

Memory

Stage 2 of fetch-execute cycle: instruction (now in IR) is decoded by control logic to determine which operation to perform

in

Control

logic

rd/wr

addr

ALU

PC

CMPUT 229

fetch execute cycle execute
Fetch-execute cycle: execute

IR

Registers

out

Stage 3 of fetch-execute cycle: CPU performs the operation (for example, arithmetic on two registers)

Memory

in

Control

logic

rd/wr

addr

ALU

PC

CMPUT 229

fetch execute cycle update pc
Fetch-execute cycle: update PC

IR

Registers

out

Stage 4 of fetch-execute cycle: PC’s value is updated to point to the next instruction

Memory

in

Control

logic

rd/wr

addr

ALU

PC

CMPUT 229

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