Digital Logic Structures

• MOS transistors

• logic gates

• functional units of a computer

MOS: Metal-oxide Semiconductor• Basic electrical circuit: power supply,

switch, lamp• manipulating the switch makes/breaks

the circuit

2 Basic Types of Transistors

• N-type: acts as a closed circuit when given a logically high voltage

• P-type: acts as a closed circuit when given a logically low voltage

Circuits with both are called CMOSCircuits with both are called CMOS

gate gate

Logic Gates

• basic logic structures (AND, OR, NOT) are created out of CMOS transistors

• inverter: recall the truth table for NOT

in outin out

0 1

1 0

in out

OR and NOR gates

• given the circuit, build the truth table

• how do we get OR?

AND and NAND gates

• Construct NAND first, just as with NOR• NAND and NOR technology very widely

used

• Inverter• AND, NAND• OR, NOR• a single bubble on an input or

output denotes an inverter• multiple-input gates

Notation for Digital Logic Gates

Expressions to Truth Tables

NOT ((NOT A) or B)

(A or B)

Expressions to Truth Tables

NOT ((NOT A) or B)

(A or B)

A A B OR C

0

0

1

1

1

1

0

0

0

1

0

1

1

1

0

1

0

0

1

0

DeMorgan’s Laws

• A AND B = A OR B

• A OR B = A AND B

Logic Structures

• we build logic structures out of logic gates

• logic structures are components of the microarchitecture of a computer

• 2 kinds of logic structures• some store information• some do not store information

Combinational Logic Structures

• output is completely determined by the combination of input values

• examples:• decoder• multiplexor (MUX)• full adder

Decoder

• outputs one 1 and the rest 0s where the 1 corresponds to a unique input pattern

• for n inputs lines, 2n output lines

• the output line that has the value 1 is asserted

MUX

• selects an input and connects it to the output

• for 2n inputs lines, n select lines

• MUX is represented by an upside down trapezoid

4 Input MUX

Single Bit Adder

a b ci co s0 0 0 0 00 0 1 0 10 1 0 0 10 1 1 1 01 0 0 0 11 0 1 1 01 1 0 1 01 1 1 1 1

4bit Adder

Logical Completeness

• we can build a circuit for any truth table using AND, OR, and NOT

• proof by construction:• draw vertical lines for all inputs• for each 1 in the output value, connect 1s in

the input directly to an AND gate (invert 0s); repeat for each row

• OR all of the AND gates together

Basic Storage Elements

• R-S latch: simple structure that stores 1 bit of information

• implemented with NAND gates• start with quiescent state (R=S=1)• as long as ‘a’ is 1, it stays 1 (same for 0)

1

1

1

0

1

1

0

1

Reset/Set the Latch

• put 0 on S while R=1, causes ‘a’ to become 1

• put 0 on R while S=1, causes ‘a’ to become 0

• behavior is undefined if both go to 0

1

1

0

1

10

1

0 1

10

quiescent

The Gated D-latch

• uses the R-S latch, plus some additional logic gates

• D is the value that is stored, but it is only set/reset when WE (write-enable) is 1

• D’s value causes one of R or S to become 0

01

0 11

10

10

01

Register• every computer offers a number of registers:

high speed special memory locations• some registers have special meaning (e.g., PC is

the program counter)• note that outputs in the 4-bit register are labeled

by Q(n-1:0)

Memory

• memory location: collection of bits, has to be uniquely identified

• identified by an address• the number of bits we have to represent

addresses determines the maximum number of locations that can be accessed in memory

224 = 16,777,216 = 16MB locations

Addressibility

• the number of bits we have to represent the data gives the addressibility of the memory. E.g. The size of each memory location.

• byte addressibility: convenient since characters are one byte

• supercomputers may be 64-bit addressible for 64-bit floating point numbers

22 by 3 Memory

Address Decoding

Logic

Gated D-Latches

3-bit Value to be stored

4 Rows4 memory Locations

• Trace how data is output and stored

Preview of the LC-2

• memory• registers

• Data (R0…R7, MDR, …)• Control (PC, MAR, ….)

• multiplexers (MUXs) • Arithmetic Logic Unit (ALU)

• Arithmetic operations (add, subtract)• Bitwise logical operations (or, and, ..)