Fundamentals of Computer Science: Data Representation and Manipulation

Representation of Audio

• Analogue Information:

  • Continuously variable data representing natural phenomena such as sight and hearing.

  • Real numbers (numbers with decimal points) are used to represent analogue information.

• Data Derived from Analogue Information:

  • Computers often handle data derived from analogue sources, including:

  • Audio (e.g. music, speech)

  • Images (e.g. diagrams, photographs)

  • Video (which includes both images and audio).

• Audio Signal Representation:

  • Audio signals can be represented as a series of numbers denoting the value of the signal at consecutive points.

• Example of Waveform Representation:

  • A waveform can be represented by a bit string, such as:

  • 0001 1001 1111 1010 0101.

• Factors Affecting Quality of Digitized Signal:

  1. Frequency:

  • Refers to the number of samples taken per second.

  1. Accuracy:

  • Denotes the number of bits used for each sample.

• Importance of Sampling Rate:

  • If the sampling rate is too slow, the representation might not accurately depict the original signal.

  • Example: A graph showing a distorted sound signal, which does not match the original.

A/D Conversion

• Analogue to Digital and Digital to Analogue Conversion:

  • Conversion between analogue signals and binary data is performed by electronic components known as A/D (Analogue to Digital) and D/A (Digital to Analogue) converters.

• Sound File Formats on Windows:

  • .wav files store audio as a simple list of data values with a defined sample rate.

  • Audio CDs contain uncompressed lists of data values with additional error-checking information.

  • Compressed formats (e.g. mp3) utilize sophisticated techniques to minimize the data required for audio representation.

Representation of Images

• Bitmap Images:

  • Digital images often consist of individual pixels and are known as bitmap images.

  • Each image is represented as an array of numbers, with each number specifying the color of one pixel.

• Vector Images:

  • An alternative representation to bitmap images is vector images, which use geometric shapes.

• Color Representation:

  • Colors are typically represented using the RGB model:

  • Red, Green, and Blue combined in various amounts.

  • Example illustrating this can be found in online resources.

Additive vs. Subtractive Color

• Additive Color Model:

  • Achieved by adding varying intensities of red, green, and blue light.

• Subtractive Color Model:

  • Applies to paints and printing processes where colors are created by subtracting varying amounts of light.

• Factors Affecting Image Data Requirements:

  • The amount of binary data necessary for image representation depends on:

  • Resolution:

    • Defined as the count of pixels the image contains.

    • Higher resolution allows for clearer images with finer details.

  • Color Depth:

    • Refers to the number of bits used to signify each pixel’s color.

    • For example, 3 bits/pixel gives 8 colors, while 24 bits/pixel provides 16 million colors, which is the standard for RGB images.

Calculating Image Size

• Example Calculation:

  • Given an image of size 800x600 pixels with a color depth of 24 bits:

  • Formula:

  • Total pixels = 800 px (height) × 600 px (width) = 480,000 pixels.

  • Total bits = 480,000 pixels × 24 bits/pixel = 11,520,000 bits.

  • Convert to bytes:

    • $ext{Total Bytes} = rac{11,520,000 ext{ bits}}{8} = 1,440,000 ext{ bytes}$

    • Convert to KB:

    • $ext{Total KB} = rac{1,440,000 ext{ bytes}}{1024} = 1,406.25 ext{ KB}$

    • Convert to MB:

    • $ext{Total MB} = rac{1,406.25 ext{ KB}}{1024} = 1.37 ext{ MB}$

    • Thus, the image requires 1.37 MB of memory space.

Manipulating Binary Data – The ALU

• Data Manipulation:

  • In computer applications, data (including numbers, text, images) is manipulated via software instructions executed by the CPU.

• Arithmetic Logic Unit (ALU):

  • The ALU within the CPU performs a variety of arithmetic and logical operations.

Binary Arithmetic

• Binary Addition:

  • Functions similarly to decimal addition, with the notion of carrying applicable in both systems.

  • Example illustrated:

    • egin{array}{cccc}
      & 1 & 1 \

    • & 1 & 0 \
      ext{(Decimal: 3 + 2)} \
      = & 1 & 0 & 1 \
      ext{(Decimal: 5)}
      \end{array}

• Carry Bit:

  • Utilized when the result of binary operations exceeds the storage capacity of CPU registers.

  • Specifically, the carry bit is kept in a separate memory location known as the carry flag.

Signed Binary Numbers – Two’s Complement Representation

• Signed Number Representation:

  • Signed numbers (capable of being positive or negative) are normally encoded as Two’s Complement.

• Two’s Complement Conversion:

  • To derive the two’s complement of a number, invert each bit and add one:

  • Example:

    • For positive 5 ($0 1 0 1$) - Invert to $1 0 1 0$, then add 1 to get $1 0 1 1$ (representing -5).

• Sign Bit Indicator:

  • The leftmost bit (most significant bit) serves as the sign bit, where:

  • ‘0’ denotes positive values.

  • ‘1’ denotes negative values.

• Three-Bit Two’s Complement Table:

  • The following is a representation of three-bit two's complement values:

  • +3: $0 1 1$

  • +2: $0 1 0$

  • +1: $0 0 1$

  • 0: $0 0 0$

  • -1: $1 1 1$

  • -2: $1 1 0$

  • -3: $1 0 1$

  • -4: $1 0 0$

• Arithmetic Operations in Two’s Complement:

  • Subtraction can be performed using addition:

  • $A - B = A + (-B)$

Arithmetic Overflow

• Understanding Overflow:

  • Due to fixed-size CPU registers, arithmetic results must fit within specified bit limits.

  • For instance, using three bits to represent values limits us to a range of -4 to +3.

  • Attempting operations outside this range can yield incorrect results, an issue known as arithmetic overflow.

• ALU Overflow Detection:

  • The ALU can recognize overflow scenarios and sets the overflow flag to ‘1’ whenever arithmetic overflow occurs.

• Real-World Example:

  • The INS (Inertial Navigation System) encountered a runtime error when converting a 64-bit number to a 16-bit format without adequacy checks, leading to malfunctioning.

Representation of Real Numbers

• Real Number Structure:

  • Real numbers consist of both whole and fractional components (e.g., 13725.00173).

  • As a simplification, a binary representation might use separate bits for whole and fractional parts, such as 32 bits each.

• Precision Representation:

  • A configuration with 7 digits for whole parts and 3 for fractions can adequately represent larger numbers but may lead to rounding issues for smaller numbers, which necessitates more significant precision.

• Floating Point Format for Real Numbers:

  • A common method to represent real numbers is using floating-point format:

  • General form: $ext{value} = ext{mantissa} imes ext{base}^{ ext{exponent}}$

Rules of Floating-Point Arithmetic

• Operations Overview:

  • Addition/Subtraction:

  • Align exponents before performing the operation on the mantissae.

  • Multiplication/Division:

  • Combine exponents and manipulate mantissae accordingly.

Floating Point Examples

• Calculating Addition and Multiplication:

  • Addition:

  • $2.6 imes 10^{3} + 1.3 imes 10^{5} = 2.6 imes 10^{3} + 130 imes 10^{3} = 132.6 imes 10^{3} ext{ or } 1.326 imes 10^{5}$

  • Multiplication:

  • $2.6 imes 10^{3} imes 1.3 imes 10^{5} = (2.6 imes 1.3) imes 10^{3+5} = 3.38 imes 10^{8}$

• Floating Point Unit (FPU):

  • Many CPUs possess an FPU to process floating-point operations, which typically require more time compared to integer operations.

Logical Operations

• 
  

• ALU Capabilities:

  • Beyond arithmetic, the ALU executes a range of logical operations including AND, OR, NOT, and XOR (exclusive OR), producing binary outputs of ‘1’ (TRUE) or ‘0’ (FALSE).

• 
  

• Logical Operations Examples:

  
  

  • 
    

  • AND:

    A	B	A.B
    0	0	0
    0	1	0
    1	0	0
    1	1	1

  • 
    

  • OR:

    A	B	A+B
    0	0	0
    0	1	1
    1	0	1
    1	1	1

  • 
    

  • NOT:

    A	NOT A
    0	1
    1	0

  • 
    

  • XOR:
    

    A	B	A+B
    0	0	0
    0	1	1
    1	0	1
    1	1	0
    Boolean Operations in High-Level Languages:
    Many languages like Python facilitate Boolean operations, as shown:
    python A = True B = False print(A and B) # Output: False
    Additional ALU Operations

    • Higher-Level Logical Operations:

      • SHIFT:

      • Moves bits in a register left or right.

      • ROTATE:

      • Similar to SHIFT but allows the displaced bits to wrap around.

      • Important for cryptographic processes.

      • COMPARE:

      • Compares contents of two registers, setting a status flag if they match.

    Summary of Key Concepts

    • Representation of audio in binary data.

    • Representation of images in binary data.

    • Binary addition and its processes.

    • Representations of negative binary numbers – two's complement.

    • Representations of real numbers in binary data.

    • Details about the ALU and its role.

    • Implications of arithmetic overflow and recognition by the overflow flag.

    • Understanding logical operations AND, OR, XOR, and NOT, along with shift and rotate operations.