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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

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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

  • 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

  • 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

  • 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


Gates

NOT

In

Out

0

1

1

0

CMPUT 229


AND

In 1

In 2

Out

0

0

0

0

1

0

1

0

0

1

1

1

Gates

CMPUT 229


Gates

NAND

In 1

In 2

Out

0

0

1

0

1

1

1

0

1

1

1

0

CMPUT 229


In 1

In 2

Out

0

0

0

0

1

1

1

0

1

1

1

1

Gates

OR

CMPUT 229


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

  • 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


CMPUT 229


Implementation of Logic Gates

  • CMOS

    • The standard, 4000 series (4011)

CMPUT 229


Implementation of Logic Gates

  • TTL

    • The 4011's TTL counterpart is the 7400

CMPUT 229


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

  • 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 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


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


  • DeMorgan’s Law

CMPUT 229


In 1

In 2

Out

0

0

0

0

1

1

1

0

1

1

1

1

CMPUT 229


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

  • A combinatorial circuit that adds binary values

    • according to this truth table

Carry is equivalent to AND

Sum is equivalent to XOR

CMPUT 229


Adder

Carry

In 1

In 2

Sum

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

CMPUT 229


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


half adders

Full 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

A3

A2

A1

A0

B3

B2

B1

B0

out3

out2

out1

out0

CMPUT 229


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 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


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

  • Computers need to do more than just addition

    • arithmetic: + – * / %

    • logic: & | ~ << >>

  • Need a circuit that can select operation to perform

CMPUT 229


Multiplexer (Mux)

B

A

A

B

S

S

C

C

CMPUT 229


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 (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


  • 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

  • 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


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 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-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-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

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


when read/write = 0, no signals in box can change, data out holds value regardless of data in

Flip-flop: reading

data in = ?

?

0

0

0

1

0

?

?

1

0

data out = 1

0

read/write = 0 (read)

CMPUT 229


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 Gates

CMPUT 229


Flip-flop using NAND Gates

Data In

Data Out

read/write

CMPUT 229


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


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

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

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


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)

  • 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


CPU

IR

Registers

out

Memory

in

Control

logic

rd/wr

addr

ALU

PC

CMPUT 229


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

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

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

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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