Number Systems and Conversions

Positional Number Systems

• The base, or radix, defines the count of unique digits used in a number system.
• The binary system (base 2) uses 0 and 1.
• The decimal system (base 10) uses 0 through 9.

Binary to Decimal Conversion

• Multiply each digit by its corresponding power of 2, then sum the products to convert binary to decimal.
• Example: $(101101)2 = (45){10}$

Decimal to Binary Conversion

• Divide the decimal number successively by 2 until the quotient is 0 to convert decimal to binary.
• The remainders, read in reverse order, form the binary equivalent.
• Example: $(45)_{10} = (101101)_2$

Octal and Hexadecimal Systems

• The octal system (base 8) uses the digits 0 through 7.
• The hexadecimal system (base 16) uses the digits 0 through 9 and the letters A through F. "A" through "F" represent 10 through 15.

Conversions Involving Octal and Hexadecimal

• To convert from binary to octal, group binary digits into sets of three from right to left.
• To convert from binary to hexadecimal, group binary digits into sets of four from right to left.
• To convert from octal/hexadecimal to binary, convert each digit to its equivalent 3 or 4-bit binary representation.

Signed Integer Representation

• Signed integer representations include sign-magnitude, 1's complement, and 2's complement.
• 2's complement is the most common method for representing signed integers in computers.

Sign-Magnitude Representation

• The leftmost bit signifies the sign: 0 for positive, 1 for negative.
• The remaining bits represent the magnitude.

1's Complement Representation

• Invert each bit of the number (0 to 1, 1 to 0) to find the 1's complement.

2's Complement Representation

• Invert all bits (1's complement) and add 1 to find the 2's complement.

Addition and Subtraction of Signed Numbers

• Use regular binary addition for adding signed numbers.
• Overflow in 2's complement addition is detected if carry-in and carry-out of the most significant bit are different.
• To subtract signed numbers, convert B to its 2's complement form and add it to A: $A - B = A + (-B)$.

Floating-Point Representation

• It is based on scientific notation and represents real numbers.
• $Number = Sign \times Mantissa \times Base^{Exponent}$

IEEE 754 Standard

• It is a common standard that represents floating-point numbers.
• Single precision (32-bit) has a 1-bit sign, 8-bit exponent, and 23-bit mantissa.
• Double precision (64-bit) has a 1-bit sign, 11-bit exponent, and 52-bit mantissa.

Components of Floating-Point Numbers

• The sign bit is 0 for positive and 1 for negative.
• The exponent is biased; bias is used to represent positive and negative exponents. Single precision bias is 127, while double precision bias is 1023.
• The mantissa is normalized to have a single non-zero digit to the left of the decimal point, where the leading 1 is implicit.

Normalization

• Move the decimal point until only one non-zero digit remains to the left and adjust the exponent accordingly.

Character Representation

• It covers standards like ASCII, Extended ASCII, and Unicode.

ASCII

• American Standard Code for Information Interchange.
• 7-bit code representing 128 characters.

Extended ASCII

• 8-bit code representing 256 characters including additional symbols and characters from different languages.

Unicode

• A comprehensive character encoding standard using variable-length encoding (UTF-8, UTF-16, UTF-32).

Image Representation

• It includes raster (bitmap) and vector graphics.

Raster Graphics (Bitmap)

• Represent images as a grid of pixels, each storing color information. Examples: BMP, JPEG, PNG, GIF.

Vector Graphics

• Uses mathematical descriptions of shapes (lines, curves, polygons) to represent images.

Color Representation

• RGB (Red, Green, Blue) defines pixel color by the intensity of red, green, and blue light.
• CMYK (Cyan, Magenta, Yellow, Black) is used in printing.

Audio Representation

• It encompasses sampling, sampling rate and bit depth

Sampling

• Analog audio signals are converted to digital by sampling at regular intervals.

Sampling Rate

• The number of samples taken per second. For example, 44.1 kHz for CD quality.

Bit Depth

• It is the number of bits used to represent each sample and determines dynamic range.

Audio Formats

• Uncompressed formats: WAV, AIFF.
• Lossless Compression formats: FLAC, ALAC.
• Lossy Compression: MP3, AAC.

Video Representation

• Topics include video encoding, compression, and interframe compression.

Video Encoding

• It's a sequence of still images (frames) displayed rapidly, measured by frames per second (fps).

Video Compression

• Codecs like H.264 (AVC), H.265 (HEVC), and VP9 are used to reduce the size of video files.

Interframe Compression

• Techniques exploit similarities between consecutive frames to reduce redundancy. Key Frames (I-frames) are complete frames, while Difference Frames (P and B frames) store differences from previous or subsequent key frames.