MCS-170 The Nature of Computer Science

Fall 2006

Lecture 13

1. Data Representation

How does a computer store data such as text, numbers, sound, video?

Before computers, data was primarily stored in Analog form

Analogous to real life, signals can vary across an infinite range of values.
Example - Wrist watch with sweep hand is an analog device, continuous read-out of time

Computers store data in Digital form - as a series of discrete values.

Example - Digital watch

Using analog to digital conversion a continuous signal (sound wave, light source) can be represented by a series of numbers, each one representing a brief sample of the signal. The numbers can then be stored as binary data.

An approximation to the original signal can then be created from the stored data.

Here is a diagram illustrating the process of digital to analog conversion

Analog Storage

Advantage: Continuous Sampling: Records a continuous stream of information about the signal, we get the most information possible about the signal
Disadvantage: Reproduction Errors: If we want to reproduce or duplicate signal repeatedly, then we lose precision (a tape of a tape of a tape is very poor in quality)

Digital Storage

Advantage: Lossless Reproduction: No matter how often we reproduce data of signal, it is always the same
Disadvantage: Sampling Errors: If we sample at too low a rate, we will lose important signal information

2. Binary Numbers

All data stored in a computer is stored in binary format -- a series of 1's and 0's

Recall: The decimal system uses the digits 0-9 to represent a number, and uses a "place holder" system

For example, the number 145 stands for the number whose value is 1*100 + 4*10 +5*1, that is, if we label columns of the number, we get
```
          H | T | O
          1 | 4 | 5
```
where"H" is the hundreds column (or place), "T" is the tens column, and "O" is the ones column.

Alternatively, we see that the ones column can be interpreted as 10^0, the tens column as 10^1, and the hundreds column as 10^2, such that the number can be written as
```
        10^2|10^1|10^0
          1 |  4 |  5 
```
so 145 is really {(1*10^2)+(4*10^1)+(5*10^0)}.

The binary system works under the same principles as the decimal system, only it operates in powers of 2 rather than powers of 10.

Instead of the columns

            10^2|10^1|10^0

             2^2|2^1|2^0

Instead of using the digits 0-9, we only use 0-1

For example, the binary number 1011 is actually the decimal number 11, since the number represents {(1*2^3) + (0*2^2) + (1*2^1)+(1*2^0)} = 8 + 2 + 1 = 11.

What decimal number is represented by 101101 ?

3. Storing Integers

To store different integers requires differing sizes of binary numbers

13 = 1011
116=1110100

Modern computers use fixed-width storage for integers - typically 32 bits, (leading 0's are put in front of the binary number to fill out all of the 32 bits)

13 = 00000000000000000000000000001011
116= 00000000000000000000000001110100

Thus, 2^32 = 4,294,967,296 bit patterns are possible. To store positive and negative numbers, we reserve about half of the patterns for positive numbers and about half for negatives.
Negative numbers are signified by setting the left-most bit to be 1. The left-most bit is not actually part of the number, but is just the equivalent of a +/- sign. "0" indicates that the number is positive, "1" indicates negative. (see Reed, page 218) This is called Two's-complement notation

Negative number with largest absolute value (2^-31) corresponds tl the smallest negative bit pattern
10000000000000000000000000000000
Negative number with smallest absolute value (-1) corresponds to the largest negative bit pattern
11111111111111111111111111111111

3. Floating Point Numbers

The real number 1234.56 can be represented as 1.23456e3 (1.23456 × 10³) in scientific notation.
In scientific notation, the numbers are normalized so only one digit appears to the left of the decimal point.
-990.0 is represented as -9.9e2 (-9.9 × 10²)
We can think of this as the pair of integers (-99,2)
Floating-point notation

Sign-bit, Exponent, Mantissa (fractional part)
IEEE single-precision floating-point representation (32 bits)

Sign bit
Exponent (eight bits)
Mantissa (remaining 23 bits)
Range of numbers: -1.75 × 10³⁸ to 3.40 × 10³⁸ (approximately)
Can get roughly 7 significant digits of precision

4. Strings

Strings are stored as sequences of characters.

A character (like "a", "#", etc) is stored as a binary number in a standard code, called the ASCII (American Standard Code for Information Interchange) code.

Overhead of Reed, page 221

Data Representation Page

5. Images

Bitmaps

A bitmap is a collection of dots, each of which is called a pixel (short for picture element).
A black-and-white pixel can be represented as a single bit.
A colored pixel is represented by a 24-bit code of three bytes giving the intensity of red, green, and blue (RGB value).
For example, the HTML color brown is represented by (165,42,42).

Image formats are usually compressed bitmaps

Common Web image formats

Graphics Interchange Format (GIF) - a lossless format
Joint Photographic Experts Group (JPEG) - a lossy format
Motion Pictures Experts Group (MPEG) for motion video
Here is an example of a GIF image - 24 bits to represent each pixel
Here is an example of a simpler gray-scale image - 8 bits for each pixel

6. Sounds

A sound wave can be represented as a waveform whose amplitude vaies with time.

With digital sampling, the waveform can be represented by a sequence of discrete values, measured at regular intervals and then stored as as sequence of numbers.

The frequency of measurement determines the fidelity of the reproduction.

8,000 samples/sec for long-distance telephony.
44,100 samples/sec for CD-quality sound.

Some widely used encoding schemes:

Musical Instrument Digital Interface (MIDI) in music synthesizers encodes what instrument is to play which note for what length of time
MP3 (MPEG1 - Layer3) is the audio compression format used in MPEG1