5 - Fundamentals of Data Representation Flashcards
What does MIDI stand for?
Musical Instrument Digital Interface
Describe the advantages of using MIDI files for representing music.
- A MIDI file is a more compact representation than an equivalent sampled recording.
- The performance data be easily manipulated, for example, it’s easy to change instruments or change the pitch or duration of a note.
- MIDI supports a very wide range of instruments providing more choices for music production.
Explain how MIDI represents music and the advantages of using MIDI for representing music instead of using sampled sound.
Music is represented as a sequence of MIDI event messages. Examples of data that might be contained in a message are channel, volume and note envelope. MIDI messages are usually two or three bytes long and 16 channels are supported.
Advantages:
• More compact representation
• Easy to modify the music, such as change the instrument
• Musical score can be generated directly from a MIDI file
How do you add two unsigned binary integers?
eg. 01001101 + 01101010
Align the binary numbers and perform column addition.
eg.0 1 0 0 1 1 0 1 0 1 1 0 1 0 1 0 + ────────── 1 0 1 1 0 1 1 1 ────────── ¹¹
How do you multiply two unsigned binary integers?
eg. 11011 x 11
Align the binary numbers and perform long multiplication.
eg. 110 11 11x ─────── 110 11 110 11 0+ ───────── 1 0 1 0 0 0 1 ───────── ¹¹¹¹¹
What is two’s complement?
A method of working with signed numbers using fixed-width binary where the most significant (left most) bit is the sign bit.
How do you convert from decimal to signed binary (using two’s complement)?
eg. -97₁₀
If the number is negative, place a 1 under the MSB then add a 1 under the column headings which, when added to the value of the MSB, give the number.
eg. -128 + 16 = -112
- 112 + 8 = -104
- 104 + 4 = -100
- 100 + 2 = -98
- 100 + 1 = -97
-128 64 32 16 8 4 2 1
1 0 0 1 1 1 1 1
-97₁₀ → 10011111₂
How do you convert from signed binary to decimal (using two’s complement)?
eg. 10011010₂
Add up the column heading with a 1 under them (where the MSB is negative).
eg. -128 64 32 16 8 4 2 1
1 0 0 1 1 0 1 0
-128 + 16 + 8 + 2 = -102
10011010₂ → -102₁₀
How do you perform binary subtraction using two’s complement?
eg. 0101₂ - 0011₂
Turn the value that is being subtracted into a negative, then add the two values.
eg. 0101₂ - 0011₂
→ 0101₂ + (-0011₂)
→ 0101₂ + 1101₂
→ (1)0010₂
a carry is generated which is ignored
0101₂ - 0011₂ → (1)0010₂
What is the range of values that can be represented using two’s complement with n bits.
the range is -2ⁿ⁻¹ to 2ⁿ⁻¹ - 1
What is fixed point binary?
Fixed point binary is a method of representing numbers with a fractional part, where the binary point is in a fixed position.
How do you convert from fixed point binary to decimal?
eg. 0110.0110₂
Set up the correct column headings according to where the point is positioned. Then add up the values of the column headings with a 1 under them.
eg. 8 4 2 1 . ½ ¼ ⅛ ¹⁄₁₆
0 1 1 0 . 0 1 1 0
4 + 2 + ¼ + ⅛ = 6.375₁₀
0110.0110₂ → 6.375₁₀
How do you convert from decimal to fixed point binary?
eg. 10.6875₁₀ (8 bits where the binary point is between the 4th and 5th bits)
eg. 10.6875 - 8 = 2.6875
2. 6875 - 2 = 0.6875
0. 8675 - ½ = 0.1875
0. 1875 - ⅛ = 0.0625
0. 0625 - ¹⁄₁₆ = 0
8 4 2 1 . ½ ¼ ⅛ ¹⁄₁₆
1 0 1 0 . 1 0 1 1
10.6875₁₀ → 1010.1011₂
How do you convert from floating point binary to decimal?
eg. 0100100100 000100
Convert the exponent and move the binary point accordingly (adding preceding 1s for negative mantissa and 0s for positive mantissa for negative exponents). Then convert to decimal.
eg. exponent → 4
0. 100100100 → 01001.00100 → 9.125
How do you convert from decimal to floating point binary?
eg. 12.125
Convert to fixed point binary move the binary point so that it’s normalised (when the first two bits are different) and convert the exponent to binary.
eg. -16 8 4 2 1 . ½ ¼ ⅛
0 1 1 0 0 . 0 0 1
01100.001 → 0.1100001
exponent = 4 → 000100
→ 01100001 000100
How do you calculate the absolute error?
approximate value - original value |
How do you calculate the relative error?
Absolute error ÷ original value
Then converted to a percentage
What are the advantages of using a normalised representation for floating point numbers?
- Maximises precision for a given number of bits
* There is a unique representation of each number
How do you normalise un-normalised floating point numbers?
eg. 00001011 0101
Move the binary point so that the first two bits are different. Then adjust the exponent to reflect the movement of the binary point.
eg. 0.0001011 → 0.1011
old exponent = 5
new exponent = 5 - 3 = 2 → 0010
→ 01011000 0010
How do you calculate the minimum amount of storage required to store a sound file?
storage requirement (in bits) = sampling rate × seconds × sample resolution
What is binary?
Binary represents numbers using 2 digits, 0 and 1. Binary is base 2.
What is hexadecimal?
Hexadecimal represents numbers using 16 digits, the numbers 0 - 9 and the letters A - F. Hexadecimal is base 16.
What is decimal?
Decimal represents numbers using 10 digits, 0 - 9. Decimal is base 10.
How do you convert from Decimal to Binary?
eg. 119₁₀
Set up the column headings and move from left to right putting a 1 under the combination of numbers that make up the decimal value.
eg. 119 - 64 = 55
55 - 32 = 23
23 - 16 = 7
7 - 4 = 3
3 - 2 = 1
1 - 1 = 0
1286432168421
01110111
119₁₀→01110111₂
How do you convert from Binary to Decimal?
eg. 11010111₂
Add up the column headings with a 1 under them.
eg. 128 64 32 16 8 4 2 1
1 1 0 1 0 1 1 1
128 + 64 + 16 + 4 + 2 + 1 = 215
11010111₂ → 215₁₀
How do you convert from Decimal to Hexadecimal?
eg. 527₁₀
Divide the decimal number by 16 (treating as integer division). Write down the remainder (in hexadecimal). Then repeat with the result of the division until the result is 0. The number in hexadecimal is the sequence of remainders in reverse order.
eg. 527 DIV 16 = 32
527 MOD 16 = 15 15 → F
32 DIV 16 = 2
32 MOD 16 = 0
2 DIV 16 = 0
2 MOD 16 = 2
256 16 1
2 0 F
527₁₀ → 20F₁₆
How do you convert from Hexadecimal to Decimal?
eg. C7F₁₆
Multiple each of the column headings by the value under them. The total is the decimal value.
eg. 256 16 1
C 7 F
256 * 12 + 16 * 7 + 1 * 15 = 3199
C7F₁₆ → 3199₁₀
How do you convert from Binary to Hexadecimal?
eg. 10111110₂
Split the binary number into nibbles and find the hexadecimal value that represents each nibble.
eg. 8 4 2 1 8 4 2 1
1 0 1 1 1 1 1 0
→ B → E
10111110₂ → BE₁₆
How do you convert from Hexadecimal to Binary?
eg. 7D₁₆
Find the binary value that represents each digit of the hexadecimal number and join them together.
eg. ↓ 7 ↓ D
8 4 2 1 8 4 2 1
0 1 1 1 1 1 0 1
7D₁₆ → 01111101₂
Why is hexadecimal used as a shorthand for binary?
- Long strings of 1s and 0s are difficult for a human to read.
- Writing numbers in hexadecimal form is less error prone than writing the same numbers in binary.
- It can be displayed in less space.
What is hexadecimal used for?
- To define memory locations.
- To represent MAC addresses.
- To define colours on web pages.
What is a bit?
A bit is the fundamental unit of information. It is an abbreviation of binary digit. A bit can be either a 0 or a 1.
What is a byte?
What is a nibble?
A byte is a group of 8 bits.
A nibble is a group of 4 bits.
How many different values can be represented with n bits.
2ⁿ different values.
What are the quantities of bytes using binary prefixes?
Kibibyte - KiB - 2¹⁰ bytes
Mebibyte - MiB - 2²⁰ bytes
Gibibyte - GiB - 2³⁰ bytes
Tebibyte - TiB - 2⁴⁰ bytes
What are the quantities of bytes using decimal prefixes?
Kilobyte - kB - 10³ bytes
Megabyte - MB - 10⁶ bytes
Gigabyte - GB - 10⁹ bytes
Terabyte - TB - 10¹² bytes
What symbol is used to represent bits?
What symbol is used to represent bytes?
b
B
