8. Digital TechnologyChapter 8.1 – Analogue and digital

signals

Binary numbers

• In ordinary arithmetic, we use the decimal system to represent number. Digits from 0 to 9 are used in this system.

•In the binary system numbers are represented using only two digits: 0 and 1.

Consider the decimal number 5037.

thousands hundreds tens unitsDigits:

In fact, we could write this number in terms of powers of 10.

5037 = 5x103 + 0x102 + 3x101 + 7x100

Binary numbers

• So, the digits of a decimal number are just the coefficients of various powers of 10. These coefficients can be digits from 0 to 9.

• The same idea applies to binary numbers. But instead of using powers of ten we will be using powers of 2. And the coefficients be the digits 0 or 1 instead the digits from 0 to 9.

• To express a decimal number in the binary system, we must write that number as a sum of powers of two with coefficients that are either 0 or 1.

Example: 5 = 1 x 22 + 0 x 21 + 1 x 20

So, the binary representation of 5 will be

52 = 101 (or 0101 if we want to represent it with four bits)

Binary numbers

12 = 1 x 23 + 1 x 22 + 0 x 21 + 0 x 20

122 = 1100

13 = 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20

132 = 1101

14 = 1 x 23 + 1 x 22 + 1 x 21 + 0 x 20

142 = 1110

…

Binary Decimal

0 0

1 1

10 2

11 3

100 4

101 5

110 6

111 7

1000 8

1001 9

1010 10

1011 11

1100 12

1101 13

1110 14

1111 15

Binary numbers

• With four-bit words (that is, four digits) we can only represent 16 number (from 0 to 15).

• For each of the four digits we have two choices: 0 or 1.

• The total number of choices is then 2 x 2 x 2 x 2 = 16.

• To represent larger number we have to increase the number of bits in the binary representation of the number.

Example:

43 = 1 x 25 + 0 x 24 + 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20

432 = 101011

Binary numbers

210 = 1024

29 = 512

28 = 256

27 = 128

26 = 64

25 = 32

24 = 16

23 = 8

22 = 4

21 = 2

20 = 1

753 = 512 + 128 + 64 + 32 + 16 + 1

So,

7532 = 1011110001

A simple way of converting a decimal number into a binary number.

Binary numbers

• Given a number in binary form, we call the first non-zero digit the most significant bit (MSB) and the last digit (the digit the number ends with) the least significant bit (LSB).

For example, 01110 has 1 as it MSB and 0 as it LSB

• The MSB is associated with the highest power of 2, and so it is the digit that mostly determines the value of the number.

How microphones work

1. Sound waves carry energy toward the microphone.

2. The diaphragm moves back and forth when sound waves hit it.

3. The coil, attached to the diaphragm, moves back and forth as well.

4. The permanent magnet produces a magnetic field that cuts through the coil. As the coil moves back and forth through the magnetic field, an electric current flows through it.

5. The electric current flows out from the microphone to an amplifier or sound recording device.

Analogue and digital signals

• When one speaks into a microphone, a voltage is created in the microphone.

• The voltage is proportional to the actual physical movement of the diaphragm of the microphone.

• A large voltage is created when the diaphragm moves fast, and a small voltage when it moves slowly.

• The voltage signal so generated varies continuously between two extreme values.

• Such signals are called analogue signals.

Analogue and digital signals

Analogue signals are continuous signals, varying between two extreme values in a way that is

proportional to the physical mechanism that created the signal.

Analogue and digital signals

A digital signals is a coded form of a signal that takes the discrete values of 0 and 1.

Analogue and digital signals

•Consider a potential divider circuit.

•The emf of the battery is 8V which means that the reading of the voltmeter can be any number between 0 and 8, depending on where the lead connects to the variable resistor R.•The signal generated in the voltmeter is an analogue signal.

Analogue and digital signals

• Imagine that the point of contact is moved from the bottom end of the resistor to the top at constant speed and assume that this is done in 4 ms.

• Then, the reading of the voltmeter would be the time-dependant signal.

Analogue and digital signals

• This analogue signal must be sampled, which means it must be measured.

• This is done at regular intervals of time.• The number of times per second the signal is sampled is

called the sampling rate or sampling frequency.• Sampling the signal means that we observe it for very

short intervals of time, wait, and then sample it again.• Thus wee do not, in general, know hoe the signal

behaves in between the instants of time when it is sampled.

• Typically, for audio signals, a sampling rate of 8000 times per second is used.

• This means that such an audio signal is sampled every 1/8000=125 s.

Analogue and digital signals

• The actual duration of one sample is very short (1.0 s or even less).

• This is why sampled signals are represented by vertical lines of practically zero width.

• When a analogue signal is converted into a digital signal, that is, when we convert a voltage into binary number, we must decide how big will our bit word be.

• If we use two-bit words, the we will have at most 22 = 4 words (00, 01, 10, 11).

• If, instead, we use three-bit words, than we will have 23 = 8 words (000, 001, 010, 011, 100, 101, 110, 111).

Analogue and digital signals

• The range of the original voltage is divided into 4 levels (if we use 2-bit words) and each level will be assigned a 2-bit word.

• In each level there is a lower boundary and an upper boundary

• In this case, there was a loss of information during the digitization of the original data.

• To improve that, we must use a higher sampling frequency and use 3-bit words or more.

Analogue and digital signals

• The process of dividing the range of the analogue signal into a set of levels is called quantization and the levels themselves are called quantization levels.

• The number of quantization levels is determined by the length of the word to be used, that is, by the number of bits used. With n bits the number of quantization levels is 2n.

• This gives rise to the notion of quantization error. Suppose that the analogue signal varies from a minimum value of m and a maximum value of M and we use n-bit words to digitize it.

• The number of quantization levels is 2n, and so at each sampling the analogue signal will take one of the 2n values.

Analogue and digital signals

• The quantityn

mMq

2

is known as the quantization error of the digitization process.

• Two analogue signals that differ less than the quantization error are assigned the same binary number.

• Obviously, the bigger the quantization error, less accurate the digital signal will be.

Analogue and digital signals

Compact disks

• The CD is a disk of diameter 12cm. The analogue signal is converted into a digital signal (‘0’s and ‘1’s) and then imprinted on the CD.

•A compact disk (CD) is a device on which information can be stored in digital form and the retrieved.

•This is done by doing marks called pits on the CD. The parts of the CD without pits are called lands.

•The edge of a pit corresponds to binary ‘1’.

•A series of pits is made along a path that spirals from the centre of the disk outwards.

Compact disks

1600 nm 830 - 3560 nm500 nm

landpit

The path has a depth of 125nm.

Compact disks

• The bottom part of the disk (the side that is actually being reads) is covered with optically transparent material (polycarbonate).

•A CD is read using a laser beam. The laser cannot have zero width.

•So when the beam is incident near the edge of a pit, a few rays will be reflected off the pit and the rest will be reflected of the land.

•This causes destructive interference and no light reaches the sensor and this corresponds to binary ‘1’.

Compact disks

Compact disks

Lands and pits and binary numbers

Compact disks

• The wavelength of the laser light used is about 780nm in air.• The refractive index of the polycarbonate material is 1.55,

which means that the wavelength of light in the polycarbonate is:

•The pit depth for destructive interference to occur must be:

nmnair 503

55.1

780

nmd 1264

503

4

Compact disks

• The laser source moves outwards and so follows the spiral of the pits and lands as the disk rotates.

• Because the circumference is getting longer as we move outwards, the rate of rotation of the disk is reduced, so that the laser can sample the disk at the same rate.

• There are clearly many technical problems to be solved here, such as stability, focusing on the right part of the spiral, and timing.

DVDs

• Because the pit length is shorter than on a CD, more data can be stored along the spiral.

• Also, data can be stored on both sides of the disk or in a double layer on the same side.

• Overall, this results in more than seven times the storage capacity compared to that of a CD.

•The digital versatile disk (DVD) is similar to the CD in many ways.

Blu-Ray

Blue laser formats have a shorter wavelength (405 nm) then CD and DVD formats which use a red laser (650 nm - DVD read wavelength). Blue ray’s blue laser beam focuses much tighter then a red laser. This allows for much tighter alignment of pits (areas of darker contrast on a recordable disc). This tighter collection of pits allows for greater storage, 27GB with the first generation of single sided blu-ray media.

LPs

• In Edison’s original sound recording in 1877, sound was incident on a diaphragm, which therefore began to vibrate.

• A needle attached to the diaphragm then made marks on a rotating tinfoil-covered cylinder.

• The ‘marks’ were a direct, mechanical copy of the actual audio signal.

• During playback, the needle retraces the pattern scratched on the cylinder surface and now makes the diaphragm move, thus reproducing the sound stored.

LPs

• In the later vinyl LPs (Long-Play) the principle of recording is essentially the same.

• But instead of a rotating cylinder, a flat rotating disk is used.

• During playback the signal is amplified electrically and fed into a loudspeaker, rather than making a diaphragm vibrate.

• LPs have a very limited storage capacity and are subject to damage by scratches and dust.

Cassettes

• These devices use magnetic recording to store data in an analogue form.

• They are called sequential devices as you must wind the cassette to get to the wanted song and this takes some time.

• The recording takes place on the ribbon of the cassette, which is made out of a strong plastic coated in ferric oxide, a ferromagnetic material.

• Ferric oxide can be permanently magnetized when exposed to a magnetic field.

Cassettes

• This current produces its own varying magnetic field.

• When the cassette is exposed to this magnetic field, a ‘copy’ of this magnetic field is created on the tape.

• During playback, the magnetic field stored on the magnetic tape will induce an electric current in a coil, which can be converted into an audio signal playing the music that was recorded.

• The advantages have been its low price and availability. Also, the tape could be erased and new material recorded

• The disadvantages refer to the sequential nature of the device, its limited storage capacity and being sensitive to high temperatures and easily damaged.

•An analogue audio signal of music can be converted to a varying electric current.

Floppy disks

• The floppy disk, like the cassette, uses magnetic recording.

• The original was invented in the mid-1960s at IBM as a way of inputting data into a computer as well as storing computer data.

• Its name comes from the flexible nature of the disk.

• Data was stored magnetically but in concentric rings, which had the advantage that one could access data on an outer ring without having to go sequentially through the intermediate data as on a cassette.

• This provided a direct access storage device.

Floppy disks

Floppy disks

Floppy Disk Drives• Qume D/T 8, 8 inch drive, 1.2 MB. This drive was made in

1980.• Tandon TM 100-2A with IBM logo, 5 ¼ inch drive, 360 KB.

This drive was made in 1983. • Sony MPF920, 3 ½ inch drive, 1.44 MB. This drive was

made in 2004.

Hard disks

• Hard disks started being used only in computers.• Nowadays, they are used in digital cameras, digital video

recorders, mobile phones and other devices.• They store data in large quantities.• The device itself consists of a number of disks made of

aluminium or glass arranged on a spindle.• The surface of the disks is covered with a material that can

me magnetized (usually cobalt).

Hard disks

• The surface may be thought to be divided into a very large number of tiny regions and each such region is the seat of a ‘0’ or a ‘1’ of digitized data.

• The growth of hard disk capacity has been exponential. Early PCs had hard disks with a capacity of just a few MB. Today’s PCs have a hard disk capacity of hundreds of GB.

• The data is stored in sectors and tracks. Tracks are concentric rings and a sector is a part of the same track.

• The data can be accessed almost instantly irrespective of its position on the disk.

Advantages of digital storage

• The capacity for data storage is huge in digital devices.

• The access to particular stored data is fast.• The retrieval of data is fast.• The storage is reliable.• The stored data can be copied or erased easily.• The stored data can be encrypted.• The data can be processed and manipulated by a

computer.• The data can be transported easily physically as

well as electronically.

Disadvantages of digital storage

• On the negative side, whereas an analogue storage system, such as ordinary photographic film, degrades slowly with time, a serious error with a digital storage device is usually catastrophic, in the sense that the data may never be recoverable.

Homework

To translate English into binary, we can use the ASCII codes where each character is given a decimal number. We write that decimal number in binary and the computer displays the character.

Work with a partner to write “Physics HL 4 life!”

in binary. The ASCII codes are on the next slide.