CSC255-702/703 CTI/DePaul1 CSC-255 Lecture 4 Negative Numbers, Images, Parity (Brookshear Chapter 1,...

CSC255-702/703 CTI/DePaul 1

CSC-255

Lecture 4Negative Numbers, Images, Parity

(Brookshear Chapter 1, pages 46-72)

Modified by Ufuk Verun from

Jim Janossy © 2002, DePaul University CTI – Chicago

and

Brookshear © 2003 Pearson Education, Inc.


Resources

This PowerPoint presentation is available for download at www.depaul.edu/~uverun/classes/CSC255/fall2002/slides/Lecture_04.ppt

Textbook: Chapter 1

Print slides at 6 slides/page, avoid waste!

Exams are based on slide content, homework and assigned readings


8-Bit Binary Numbers

So far we used no sign; i.e., all positive

11111111255

000000000

Highest 8-bit pure binary value

Lowest 8-bit pure binary value


How To Store Negative Numbers?

We can “trade” some of the number range capacity of binary numbers to represent negative integers

For example, 8 bits can store -128 to +127 rather than 0 to 255

Two’s complement and excess notation are two ways to do this


Two’s Complement System

Takes half of the bit patterns and uses them for negative numbers

Provides computational advantage: addition and subtraction are the same operation!

Most commonly used today with 32-bit binary numbers


Two’s Complement


Two’s Complement Example

Using only 3 bits to simplify things… Pure binary Two’s complement

1 1 1 7 1 1 1 -1 0 1 1 3 1 1 0 6 1 1 0 -2 0 1 0 2 1 0 1 5 1 0 1 -3 0 0 1 1 1 0 0 4 1 0 0 -4 0 0 0 0 0 1 1 3 0 1 1 3 1 1 1 -1 0 1 0 2 0 1 0 2 1 1 0 -2 0 0 1 1 0 0 1 1 1 0 1 -3 0 0 0 0 0 0 0 0 1 0 0 -4


Two’s Complement Signs

Leftmost bit indicates sign 0:+, 1:- Pure binary Two’s complement

1 1 1 7 1 1 1 -1 0 1 1 3 1 1 0 6 1 1 0 -2 0 1 0 2 1 0 1 5 1 0 1 -3 0 0 1 1 1 0 0 4 1 0 0 -4 0 0 0 0 0 1 1 3 0 1 1 3 1 1 1 -1 0 1 0 2 0 1 0 2 1 1 0 -2 0 0 1 1 0 0 1 1 1 0 1 -3 0 0 0 0 0 0 0 0 1 0 0 -4


The Two’s Complement “Forming” Rule

Take a positive binary value

Copy from right to left unchanged until you pass the first “1” encountered

After the first “1”, continue copying right to left but write the “complement” (0 for 1, 1 for 0)


Example: Finding Two’s Complement


Example: Forming Two’s Complement for -26

What is two’s complement code for value -26 using 8 bits?

+26: 0 0 0 1 1 0 1 0

-26: 1 1 1 0 0 1 1 0

Complement remaining bits


Exercise: Two’s Complement for -117

What is two’s complement code for value -117 using 8 bits?

+117: ? ? ? ? ? ? ? ?

-117: ? ? ? ? ? ? ? ?


Exercise: Two’s Complement for -210

What is two’s complement code for -210 using 8 bits? (hint: trick question)

+210: ? ? ? ? ? ? ? ?

-210: ? ? ? ? ? ? ? ?


“Decoding” Rule for Two’s Complement Values

If the sign bit is 0, the number is positive. You can read it as if it were pure binary.

Example: 0 0 0 1 0 1 0 1 = decimal 21


The “Decoding” Rule for Two’s Complement Values

If the sign bit 1, the number is negative. You must:

1. Copy bits right to left unchanged until you pass through the first “1”

2. Continue copying right to left but write the “complement” (0 for 1, 1 for 0)

3. Interpret the resulting binary value and put “-” in front of it


Example: Decoding a Two’s Complement Value

What is the decimal value of the following two’s complement binary number?

0 0 0 0 1 1 0 0

Leading digit is 0, so it’s positive.

Convert it from binary to decimal:Value is 12Make sure you know why!


Example: Decoding Two’s Complement Value

In two’s complement system, what is the decimal value of

1 1 1 1 1 0 1 0 ?

Leading digit is 1, so it’s negative. You must decode it, then convert it from

binary to decimal...


Example: Decoding a Two’s Complement Value

Decode it:

1 1 1 1 1 0 1 0

0 0 0 0 0 1 1 0 This is 6

- 6

Complement remaining bits


Exercise: Decoding a Two’s Complement Value

What is the decimal value of 10010101 in two’s complement system?


Exercise: Ranges for Two’s Complement Numbers

What is the maximum and minimum numbers that can be represented using 16-bit two’s complement system?


Exercise: Ranges for Two’s Complement Numbers

What is the maximum and minimum numbers that can be represented in 16-bit two’s complement system?Answer:Two’s complement divides the 16-bit range into two:

-215 to -1Minimum: 1000000000000000 = -32768

0 to 215 – 1Maximum: 0111111111111111= +32767


Common Two’s Complement Mistakes

This value is -6 in two’s complement:

1 1 1 1 1 0 1 0

It is incorrect to write -6 as -11111010It is incorrect to write -6 as -00000110It is incorrect to write -6 as 10000110The fact that 11111010 is negative is contained entirely in the leading “1” and two’s complement notation


Addition in Two’s Complement Notation


Overflow problem

Overflow happens when a resultant value of a mathematical operation falls outside of the range that can be represented with a given number of bits

Example: In pure binary system, 254 + 6 produces 260, but 8 bits cannot hold 260

Small values can accumulate to produce large values and cause overflow!


Overflow in Pure Binary

Add 134 and 130: 1 1 1

+ 134: 1 0 0 0 0 1 1 0+ 130: 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 = +8 ??? O v e r f l o w ! This bit is lost - The answer is

wrong!


Subtraction in Two’s Complement...

Computing 6 - 2 is same as 6 + (-2): 1 1 1 1 1 1 1

+ 6: 0 0 0 0 0 1 1 0 - 2: 1 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 = +4 Ignore overflow. Addition of positive and

negative numbers will set the sign bit correctly. In this case, 0:positive.


Subtraction in Two’s Complement...

Computing -112 - 112: 1 1 1 1

-112: 1 0 0 1 0 0 0 0-112: 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0

Cannot ignore overflow. Addition of two negative numbers should set the sign bit to 1. In this case, 0:positive, which is not correct.


Addition in Two’s Complement...

Computing 112 + 112: 1 1 1

+112: 0 1 1 1 0 0 0 0+112: 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0

Addition of two positive numbers should set the sign bit to 0. In this case, 1:negative, which is not correct.


Exercise: Addition in Two’s Complement...

Compute 68 – 120 in 8-bit two’s complement representation

+68: ? ? ? ? ? ? ? ?-120: ? ? ? ? ? ? ? ?

? ? ? ? ? ? ? ?

Is your answer correct, or did it result in overflow? How do you know? Explain.


Arithmetic Operations in Two’s Complement System

Subtraction is handled by addition

Multiplication can be done by repeated addition

Division can be done by repeated subtractionSo the same circuitry used to perform two’s complement addition can handle all four arithmetic operations!


Excess Notation

A different way to reassign half the range of bit patterns to hold negative numbersFirst pattern with high order “1” represents value 0, next highest represents value 1, and so onValues lower represent -1, -2, -3, . . .


Example: 4-bit Excess Notation


Excess Notation Example

Using only 3 bits to simplify things… Pure binary Excess notation

1 1 1 7 1 1 1 3 1 1 0 6 1 1 0 2 1 0 1 5 1 0 1 1 1 0 0 4 1 0 0 0 0 1 1 3 0 1 1 -1 0 1 0 2 0 1 0 -2 0 0 1 1 0 0 1 -3 0 0 0 0 0 0 0 -4


Excess Notation…

Named “excess” because pattern in binary is always a fixed number greater than the interpretation of the same pattern in excess notation

Excess does not offer the computational advantage that two’s complement does

Excess is used in floating point notation


Floating Point Notation

Binary and two’s complement are good for integers (whole numbers) only:(1, 2, 17, 3753647)

Floating point is used for numbers with fractional parts (3.1416, 16.53798)

Think of floating point associated with scientific notation, like 6.28462 x 107


Example: Floating Point Notation

Bits to be used for number storage are divided into three groups:

0 1 1 0 1 0 1 1

sign bit mantissa

exponent


Example: Floating Point…

0 1 1 0 1 0 1 1

Sign bit: 0 = positive, 1 = negative

Mantissa: pure binary value with assumed radix point at left; eg. .1011

Exponent: excess notation, 110 = 2, how many places right (+) or left (-) to move the radix: 10.11 = 2 + 1/2 + 1/4


Exercise: Floating Point

What is the decimal value of the following 8-bit floating point binary number? (3 bits exponent, 4 bits mantissa)

0 0 1 0 1 1 0 1


Exercise: Floating Point

What is the decimal value of the following 16-bit floating point binary number? (7 bits exponent, 8 bits mantissa)

1 1 0 1 0 1 1 1 1 1 0 1 1 0 0 1


Imprecision in Number Storage

Truncation: not enough bits to store number, so least significant digits lostNonterminating expansions like 1/3 = .3333…More nonterminating expansions exist in binary than in decimalUsers are generally shielded from these effects by application software


Example: Floating Point Representation Using 8 Bits

What is sign bit, what is exponent?


Images


Image as Bitmap (.BMP)

Image is collection of dots (pixels)Pixel = “picture element”Black & white: one bit per pixelColor: each pixel represented by combination of red, green, blue (RGB) in varying intensity, to form all colors. Three bytes per pixel: one byte for each color intensity, 0-255 value


.BMP

With 3 bytes/pixel, how many bytes are needed for following resolutions? 2400 x 1800 = 4.3 M x 3b = 13 Mb 1600 x 1200 = 1.9 M x 3b = 5.8 Mb 1280 x 960 = 1.3 M x 3b = 3.9 Mb 800 x 600 = 480 K x 3b = 1.44 Mb 640 x 480 = 307 K x 3b = 921 Kb 320 x 240 = 76.8 K x 3b = 230 Kb


Problems with Bitmaps

Bitmap image cannot be enlarged, since pixels get bigger and image gets grainy or “blocky”

.GIF and .JPG formats compress images

Raw bitmaps are not efficient, .GIF and .JPG are


GIF (.GIF) Image

Graphics Interchange Format

Developed by Compuserve (an ISP)

Stores only 256 colors

Loses some picture quality but is simple and fast

Common in computer action games


JPEG (.JPG) Image

Joint Photographic Experts Group

Stores differences between adjacent pixels, not absolute values

Uses variable-length data (values take a minimum number of bits to store)

On the average, uses only 5% of the space of bitmaps


.JPG

With 3 bytes/pixel, how many bytes are needed for the following resolutions? 2400 x 1800 = 4.3 M x 3b = 13 Mb 650K 1600 x 1200 = 1.9 M x 3b = 5.8 Mb 290K 1280 x 960 = 1.3 M x 3b = 3.9 Mb 195K 800 x 600 = 480 K x 3b = 1.44 Mb 72K 640 x 480 = 307 K x 3b = 921 Kb 46K 320 x 240 = 76.8 K x 3b = 230 Kb 12K


Vector Images

Pixels are not mapped

Equations for the lines and curves making up the image are stored

Image is stored as the instructions for drawing the image

Images are easily scaled


Vector Images…

Modern type fonts are vector images

Used in computer aided design (CAD) systems for “blueprint” drawings

Good for three-dimensional drawings

Windows metafile (.wmf) or Visio (.vsd)

Cannot produce photographic images


Error Detection and Correction


Bit Errors

Data storage mediums like memory, magnetic disk, CD/ROM can introduce errors (bits can get “flipped”)Noisy communication circuits can do this tooTechniques are used to overcome these limitations and make data storage and communication reliable


Detecting Bit Error

Parity bit: an extra bit used to detect error

Even Parity: Parity bit is set to make number of 1’s even; Example: 1 0 0 1 0 0 1 0 1

Odd parity system is identical except that bit is set to make number of 1’s odd

Receiving circuitry can detect if one bit has been flipped


Example: Generating Odd Parity Value


Memory with Parity

Some memory systems use 9 bits for each 8-bit byte

Extra bit is for parity checking

Hardware checks parity periodically behind the scenes

Your programs don’t “see” the extra bit


Variations on Parity Checks

Checkbytes or checksums - “hash” total computed when data is stored, recomputed when data is read

Cyclic redundancy codes (CRC)


Error Correction

Hamming codes (named for inventor, R.W. Hamming) - can be used to correct as well as detect some flawed bits, based on algebraic coding theory


Example: Hamming Code for Symbols A through H


Example: 1-Bit Error Correction for Pattern 010100

Calculate the number of bits different between each code on table and the given pattern Distance between A(000000) and 010100= 2 Distance between B(001111) and 010100= 4 Distance between C(010011) and 010100= 3 Distance between D(011100) and 010100 = 1 Distance between E(100110) and 010100= 3 Distance between F(101001) and 010100 = 5 Distance between G(110101) and 010100 = 2 Distance between H(111010) and 010100 = 4


Example: 1-Bit Error Correction for Pattern 010100

Based on 1-bit error correction, D is the pattern

CSC255-702/703 CTI/DePaul1 CSC-255 Lecture 4 Negative Numbers, Images, Parity (Brookshear Chapter 1,...

Documents

Transcript of CSC255-702/703 CTI/DePaul1 CSC-255 Lecture 4 Negative Numbers, Images, Parity (Brookshear Chapter 1,...