1. ECE 2110: Introduction to Digital Systems
Review #1
(Chapter 1 ,2)

2. Chapters 1,2 summary
Analog<-->Digital: 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/subtraction
Signed numbers: representations, conversions, addition/subtraction
Sign extension
Overflow (in 2’s complement operations)
Binary Codes for decimal numbers: BCD (8421) ,2421, 6311,….
Gray code
Other codes (7-bit ASCII)

3. 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.

4. Electronic and software aspect of digital design
Digital abstraction
Noise Margin
specification
Examples of software tools

5. 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: gates
LSI: ,000
VLSI: >100,000. Reached 50million in 1999

6. 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

7. 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

8. Positional Notation
Base
Radix
Weight
Radix 2, 8, 10,16

9. Unsigned numbers Conversions:
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,…

10. Hex to Binary, Binary to Hex
A2F16 = =
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 = = 5116
Padded with a zero

11. 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
= 1x23 + 0x22 + 1x21 + 1x20 + 1x x = =
Hex to Decimal
A2F16 = 10x x x = 10 x x x = = 2607

12. 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.

13. Conversion of Decimal Integer To ANY Base Example
Convert 53 to binary (R=2)
53/2 = 26, rem = 1 26/2 = 13, rem = 0 13/2 = 6 , rem = /2 = 3, rem = 0 3/2 = 1, rem = 1 1/2 = 0, rem = 1 Check =
= 1x25 + 1x24 + 0x23 + 1x22 + 0x21 + 1x = = 53 √
Least Significant Digit
Most Significant Digit

14. Binary addition/subtraction rules
cin or bin X Y
Cout s
Bout d 1

15. Binary, Decimal addition
From LSB to MSB: 1+1 = 0, carry of 1 1 (carry)+1+0 = 0, carry of 1 1 (carry) = 1, no carry = = = 1 answer = 34
from LSD to MSD: 7+4 = 1; with carry out of 1 to next column
1 (carry) = 5. answer = 51.

16. Subtraction Binary Decimal 900 1002
= 9; with borrow of 1 from next column 0 -1 (borrow) - 0 = 9, with borrow of (borrow) - 0 = 8. Answer = 899.
1002
= 1; with borrow of 1 from next column 0 -1 (borrow) - 0 = 1, with borrow of (borrow) - 0 = 0. Answer = 0112.

17. 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

18. 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.

19. 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 )
1’s complement 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)

20. 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.

21. 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.

22. 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.

23. 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.

24. 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

25. 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.

26. 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

27. 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.

28. 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 i+1 of the corresponding binary code words are the same
1: otherwise