1. Digital Circuits and Logic
Gates
Combinatorial logic
adder
ALU
Sequential logic
flip-flop
memory
CPU
fetch-decode-execute cycle
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2. 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
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3. Logic Level one (1) / zero (0)
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
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4. 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
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5. Gates
NOT In
Out 1 1
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6. Gates 1
Out
In 2
In 1
AND
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7. Gates
NAND
In 1
In 2
Out 1 1 1 1 1 1 1
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8. Gates OR 1
Out
In 2
In 1
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9. Gates 1
Out
In 2
In 1
XOR
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10. 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
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12. Implementation of Logic Gates
CMOS
The standard, 4000 series (4011)
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13. Implementation of Logic Gates
TTL
The 4011's TTL counterpart is the 7400
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14. Logic Circuits Combinatorial Logic Sequential 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
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15. 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
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16. 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
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17. are equivalent; each can be converted to the other two.
Combinatorial logic
boolean expression
X = (A & ~B) | (B & C)
truth table
logic circuit A B C X 1 A B C X 1 1 1 1 1 1
all three of forms
are equivalent; each can be converted to the other two.
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18. DeMorgan’s Law
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19. 1
Out
In 2
In 1
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20. 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
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21. Adder A combinatorial circuit that adds binary values
according to this truth table
In 1
In 2
Sum
Carry 1
Carry is equivalent to AND
Sum is equivalent to XOR
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22. This circuit can add two binary digits and is called a half-adder
Carry
In 1
In 2
Sum
This circuit can add two binary digits and is called a half-adder
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23. Adder 5 3 8 1 4 + 9 1 Why “half-adder”. . . ? 1
. . . because to add multi-digit numbers each column requires two additions
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24. Full adder
in 1
This circuit is called a full adder and can add three binary digits
in 2
half adders
carry out
carry in
This OR gate combines the carries from the two half-adders
sum
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25. Ripple-carry adder A3 A2 A1 A0 B3 B2 B1 B0 out3 out2 out1 out0
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26. Ripple-carry adder 0101 + 0110 = 1011 A3 A2 A1 A0 B3 B2 B1 B0 out31 1 1 1 1 1
out3
out2
out1
out0
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27. 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
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28. 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
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29. Arithmetic Computers need to do more than just addition
logic: & | ~ << >>
Need a circuit that can select operation to perform
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30. Multiplexer (Mux) A B A B S S C C
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31. Arithmetic Logic Unit (ALU)B
more operations here
. . .
op 0
op 1
op 2
op 3 + *
&
<<
Multiplexer: a combinatorial circuit which selects exactly one input
MUX 1 2 3 .. op
op selects operation:
0 = add, 1 = multiply, ...
out
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32. Arithmetic Logic Unit (ALU)
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
MUX 1 2 3 ..
op = 3
out = 60
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33. 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
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34. 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
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35. 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
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36. Flip-flop Flip-flop needs two operation modes Also need
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
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37. 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
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38. Flip-flop NOR gate: OR gate followed by NOT gate data in data out
read/write
read/write control: 0 = read, 1 = write
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39. Flip-flop: writing Try changing data in to 0 and watch data out1 1 1 1 1 1
data out = 1 1
when read/write = 1, data out = data in
read/write = 1 (write)
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40. Flip-flop: reading data in = ? data out = 1 read/write = 0 (read)
when read/write = 0, no signals in box can change, data out holds value regardless of data in 1 ? ? 1
data out = 1
read/write = 0 (read)
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41. Flip-flops are often drawn like this in block diagrams
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)
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42. Flip-flop using NAND Gates
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43. Flip-flop using NAND Gates
Data In
Data Out
read/write
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44. 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)
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45. Memory Decoder: feeds input to selected output, 0 to all others DEC
data in D Q
address 0 CK D Q
data out
address 1
rd/wr 1 CK 1
DEC
MUX 2 2 3 D Q 3
...
address 2
... CK D Q
address
address 3 CK
. . . millions more flip-flops . . .
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46. Memory: writing Writing value 1 to flip-flop at address 2 DEC MUX 1
data in ? D Q 1
address 0 CK ? D Q
data out
address 1
rd/wr 1 CK 1
DEC
MUX 2 2 1 3 1 D Q 3
... 1
address 2
... 1 CK ? D Q
address
address 3 CK 2
. . . millions more flip-flops . . .
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47. Memory: reading Reading value from flip-flop at address 2 DEC MUX 1 1
data in D Q
address 0 CK D Q
data out
address 1
rd/wr 1 CK 1
DEC
MUX 2 2 1 3 1 D Q 3
... 1
address 2
... CK D Q
address
address 3 CK 2
. . . millions more flip-flops . . .
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48. 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
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49. 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
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50. CPU out Registers Memory in Control logic rd/wr addr IR ALU PC
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51. Fetch-execute cycle: fetchIR
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
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52. Fetch-execute cycle: decodeIR
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
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53. Fetch-execute cycle: executeIR
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
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54. Fetch-execute cycle: update PCIR
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
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