ECE 2110: Introduction to Digital Systems

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ECE 2110 Introduction to Digital Systems

• Review 1
• (Chapter 1 ,2)

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Chapters 1,2 summary

• Analoglt--gtDigital advantages, electronic
  aspects, software aspects, digital design levels.
• IC wafer, die, classifications
  (SSI,MSI,LSI,VLSI)
• Positional number systems binary, octal,
  decimal, hex
• Unsigned numbers conversions, addition/subtractio
  n
• Signed numbers representations, conversions,
  addition/subtraction
• Sign extension
• Overflow (in 2s complement operations)
• Binary Codes for decimal numbers BCD (8421)
  ,2421, 6311,.
• Gray code
• Other codes (7-bit ASCII)

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Digital Design Basics

• Analog vs. Digital
• Why we need digital?
• Reproducibility, economy, programmability
• Digital Devices
• Gates, FFs
• Combinational output depends only on the current
  input combination
• Sequential circuits output depends on current
  input as well as past inputs. Has memory of past
  events.

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Electronic and software aspect of digital design

• Digital abstraction
• Noise Margin
• specification
• 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 gt100,000. Reached 50million in 1999

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Digital Design Levels

• Many representations of digital logic
• Device Physics and IC manufacturing
• Moores Law 1965, Gordon Moore
• Transistor level ---gtLogic 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

• Base
• Radix
• Weight
• Radix 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 lt-----gtbinary
• Octal lt-----gt binary
• (padded with zero)
• Any base lt-----gtdecimal
• Operations (binary) addition, subtraction,
  multiplication,

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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.
Binary to Decimal
1011.112 1x23 0x22 1x21 1x20 1x2-1
1x2-2 8 0
2 1 0.5 0.25
11.75
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 (R2)
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 Check 5310 1101012 1x25
1x24 0x23 1x22 0x21 1x20 32 16
0 4 0 1 53 v
Least Significant Digit
Most Significant Digit
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Binary addition/subtraction rules
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Binary, Decimal addition
Binary
Decimal
1010112 0000012---------------
1011002From LSB to MSB11 0, carry of 11
(carry)10 0, carry of 11 (carry)0 0 1,
no carry1 0 10 0 0 1 0 1 answer
1011002
34 17------ 51from LSD to MSD74
1 with carry out of 1 to next column 1 (carry)
3 1 5.answer 51.
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Subtraction
Binary
Decimal
900 - 001------- 8990-1 9 with
borrow of 1 from next column0 -1 (borrow) - 0
9, with borrow of 1 9 - 1 (borrow) - 0
8.Answer 899.
1002 - 0012 -------
01120-1 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
• Ones-Complement
• Radix-Complement Representation
• Twos-Complement

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NOTE

• Fix number of digits
• SM, 1s complement, 2s complement may be
  different for NEGATIVE numbers, but
• for positive numbers, the representations in SM,
  1s complement, 2s complement are the SAME,
  equals to the unsigned binary representation.

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Ranges (N bits)
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 )
1s complement can represent the signed integers
-(2(N-1) - 1) to
(2(N-1) - 1 )
2s 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,1s,2s complement)
• Take whatever the SIGN BIT is, and extend it to
  the left.

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Conversions for signed numbers

• Hex---gtsigned 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 ----gthex
• 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 Zero If 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
OneIf 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

• Twos-complement
• Addition rules
• Subtraction rules
• Overflow
• Out of range
• Detecting unsigned overflow (carry out of MSB)
• Detecting 2s complement overflow

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Detecting Twos Complement Overflow
Twos complement overflow occurs is Add two
POSITIVE numbers and get a NEGATIVE result
Add two NEGATIVE numbers and get a POSITIVE
resultWe CANNOT get twos 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 twos
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 (n1)-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 Code

• The 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 i1 of the corresponding binary
  code words are the same
• 1 otherwise