ECE 3110: Introduction to Digital Systems

Review #1

(Chapter 1 ,2)

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Chapters 1,2 summary

Analog<-->Digital: advantages, electronic aspects, software aspects, digital design levels.

IC: wafer, die, classifications, PLC,ASIC Positional number systems: binary, octal, decimal,hex Unsigned numbers: conversions,

addition/subtraction/multiplication/division Signed numbers: representations, conversions, addition/subtraction Sign extension Overflow Binary Codes for decimal numbers:BCD,2421,excess-3…. Gray code Other codes

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Digital Design Basics

Analog vs. DigitalWhy we need digital?

Reproducibility, economy, programmability…

Digital Devices Gates, FFs Combinational, sequential circuits

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Electronic and software aspect of digital design

Digital abstractionNoise Marginspecification

Examples of software tools

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Integrated Circuits (IC)

A collection of one or more gates fabricated on a single silicon chip. Wafer, die

Small-scale integration (SSI): 1-20 DIP: dual in-line-pin package Pin diagram, pinout

MSI: 20-200 gates LSI: 200-200,000

VLSI: >100,000, 50million (1999)

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Programmable Logic Device (PLD)

ICs which Logic function can be programmed after manufacture.

2-level AND-OR gates using user-programmable connections PLAs: programmable Logic Arrarys. PALs: programmable array logic devices Programmable logic devices (PLDs)

CPLD: complex PLD FPGA: Field-Programmable Gate Array

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Application-Specific ICs (ASIC)

Semi-custom IC: chips designed for a particular, limited product/application

Reducing chip count, size, power consumption Higher performance. NRE: nonrecurring engineering cost

Standard cells: library Gate array: an IC with internal structure as an array of

gates, unspecified interconnection

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Digital Design Levels

Many representations of digital logic Device Physics and IC manufacturing

Moore’s Law [1965, Gordon Moore]:

Transistor level --->Logic design, functional building blocks

Transistor level, Truth table, gate level logic, prepackaged blocks, equations, HDL

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Binary Representation

The basis of all digital data is binary representation. Binary - means ‘two’

1, 0 True, False Hot, Cold On, Off

We must be able to handle more than just values for real world problems 1, 0, 56 True, False, Maybe Hot, Cold, Warm, Cool On, Off, Leaky

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Positional Notation

BaseRadixWeightRadix 2, 8, 10,16

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Unsigned numbers N binary digits (N bits) can represent unsigned integers from 0 to 2N-1.

Conversions: Hex <----->binary Octal <-----> binary

(padded with zero) Any base <----->decimal

Operations (binary): addition/subtraction/multiplication/division

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Hex to Binary, Binary to Hex

A2F16 = 1010 0010 11112

34516 = 0011 0100 01012

Binary to Hex is just the opposite, create groups of 4 bits starting with least significant bits. If last group does not have 4 bits, then pad with zeros for unsigned numbers.

10100012 = 0101 00012 = 5116

Padded with a zero

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Conversion of Any Base to Decimal

Converting from ANY base to decimal is done by multiplying each digit by its weight and summing.

1011.112 = 1x23 + 0x22 + 1x21 + 1x20 + 1x2-1 + 1x2-2

= 8 + 0 + 2 + 1 + 0.5 + 0.25 = 11.75

Binary to Decimal

Hex to Decimal

A2F16 = 10x162 + 2x161 + 15x160 = 10 x 256 + 2 x 16 + 15 x 1 = 2560 + 32 + 15 = 2607

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Conversion of Decimal Integer To ANY Base

Divide Number N by base R until quotient is 0.

Remainder at EACH step is a digit in base R, from Least Significant digit to Most significant digit.

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Conversion of Decimal Integer To ANY BaseExample

Convert 53 to binary

53/2 = 26, rem = 1 26/2 = 13, rem = 0 13/2 = 6 , rem = 1 6 /2 = 3, rem = 0 3/2 = 1, rem = 1 1/2 = 0, rem = 1

5310 = 1101012

= 1x25 + 1x24 + 0x23 + 1x22 + 0x21 + 1x20

= 32 + 16 + 0 + 4 + 0 + 1 = 53

Least Significant Digit

Most Significant Digit

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cin or bin X Y Cout s Bout d

0 0 0 0 0 0 0

0 0 1 0 1 1 1

0 1 0 0 1 0 1

0 1 1 1 0 0 0

1 0 0 0 1 1 1

1 0 1 1 0 1 0

1 1 0 1 0 0 0

1 1 1 1 1 1 1

Binary addition/subtraction rules

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Binary, Decimal addition

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+ 17------ 51from LSD to MSD:7+4 = 1; with carry out of 1 to next column

1 (carry) + 3 + 1 = 5.answer = 51.

Decimal 1010112

+ 0000012

--------------- 1011002

From LSB to MSB:1+1 = 0, carry of 11 (carry)+1+0 = 0, carry of 11 (carry)+0 + 0 = 1, no carry1 +0 = 10 + 0 = 0 1 + 0 = 1 answer = 1011002

Binary

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Subtraction

Decimal

900

- 001------- 899

0-1 = 9; with borrow of 1 from next column0 -1 (borrow) - 0 = 9, with borrow of 1 9 - 1 (borrow) - 0 = 8.Answer = 899.

Binary

1002

- 0012

------- 0112

0-1 = 1; with borrow of 1 from next column0 -1 (borrow) - 0 = 1, with borrow of 1 1 - 1 (borrow) - 0 = 0.Answer = 0112.

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Representation of Negative Numbers

Signed-Magnitude Representation: Negates a number by changing its sign.

Complement Number Systems: negates a number by taking its complement. Diminished Radix-Complement Representation

One’s-Complement Radix-Complement Representation

Two’s-Complement

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NOTE:

Fix number of digits

SM, 1’s complement, 2’s complement may be different for NEGATIVE numbers, but

for positive numbers, the representations in SM, 1’s complement, 2’s complement are the SAME, equals to the unsigned binary representation.

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Ranges (N bits)

1’s complement can represent the signed integers

-(2(N-1) - 1) to + (2(N-1) - 1 )

unsigned binary can represent unsigned integers from 0 to 2N-1.

SM can represent the signed integers

-(2(N-1) - 1) to + (2(N-1) - 1 )

2’s complement can represent the signed integers

-2(N-1) to + (2(N-1) - 1)

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Sign extension

For unsigned binary, Just add zeros to the left.

For signed binary (SM,1’s,2’s complement…): Take whatever the SIGN BIT is, and extend

it to the left.

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Conversions for signed numbers

Hex--->signed decimal Given a Hex number, and you are told to convert to a signed integer (either as

signed magnitude, 1s complement, 2s complement) Step 1: Determine the sign Step 2: determine magnitude Step 3: combine sign and magnitude

Signed decimal ---->hex Step 1: Know what format you are converting to!!! Step 2: Ignore the sign, convert the magnitude of the number to binary. Step 3 (positive decimal number): If the decimal number was positive,

then you are finished no matter what the format is! Step 3 (negative decimal number): more work need to do.

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Hex to Signed Decimal Conversion Rules

STEP 1: Determine the sign! If the Most Significant Bit is zero, the sign is positive. If the MSB is one, the sign is negative. This is true for ALL THREE representations: SM, 1s complement, 2s complement.

STEP 2 (positive sign): If the sign is POSITIVE, then just convert the hex value to decimal. The representation is the same for SM, 1s complement, 2s complement.STEP 2 (negative sign): If the sign is Negative, then need to compute the magnitude of the number.

If the number is SM format, set Sign bit to ZeroIf the number is 1s complement, complement each bit.If the number is 2s complement, complement and add one.

STEP 3 : Just combine the sign and magnitude to get the result.

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Signed Decimal to Hex conversion

Step 1: Know what format you are converting to!!! You must know if you are converting the signed decimal to SM, 1s complement, or 2s complement.Step 2: Ignore the sign, convert the magnitude of the number to binary.

Step 3 (positive decimal number): If the decimal number was positive, then you are finished no matter what the format is!

Step 3 (negative decimal number): Need to do more if decimal number was negative.

If converting to SM format, set Sign bit to One:If converting to 1s complement, complement each bit. If converting to 2s complement, complement each bit and add 1.

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signed addition/subtraction

Two’s-complement Addition rules Subtraction rules

Overflow: Out of range Detecting unsigned overflow (carry out of MSB) Detecting 2’s complement overflow

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Detecting Two’s Complement Overflow

Two’s complement overflow occurs is:

Add two POSITIVE numbers and get a NEGATIVE result Add two NEGATIVE numbers and get a POSITIVE result

We CANNOT get two’s complement overflow if I add a NEGATIVE and a POSITIVE number together.

The Carry out of the Most Significant Bit means nothing if the numbers are two’s complement numbers.

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Codes

Code: A set of n-bit strings in which different bit strings represent different numbers or other things.

Code word: a particular combination of n-bit values N-bit strings at most contain 2n valid code words.

To represent 10 decimal digits, at least need 4 bits. Excessive ways to choose ten 4-bit words. Some

common codes: BCD: Binary-coded decimal, also known as 8421 code Excess-3 2421…

Codes can be used to represent numerical numbers, nonnumeric texts, events/actions/states/conditions

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How to construct Gray Code

Recursively A 1-bit Gray Code has 2 code words, 0, 1 The first 2n code words of an (n+1)-bit Gray

code equal the code words of an n-bit Gray Code, written in order with a leading 0 appended.

The last 2n code words equal the code words of an n-bit Gray Code, but written in reverse order with a leading 1 appended.

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Another method to construct Gray CodeThe bits of an n-bit binary or Gray-code

word are numbered from right to left, from 0 to n-1

Bit i of a Gray code word is 0 if bits i and i+1 of the corresponding binary

code words are the same 1: otherwise