Digital to Analog

• Many carrier facilities are analog• Many transmission media are also analog

(microwave, radio)• We can carry digital values over analog signals• We must ‘encode’ the digital data over the digital

signal• This is what is done with modems

Modemdigital Analog

Modem = Modulation/Demodulation

Encoding Digital Data

• We can encode data (bits) onto regular analog sine waves or example.

• Consider a sine wave y = sin(x)

Analog Signal

• Beyond a basic period, the wave repeats itself• The time for one non repeating part of the signal

is called the period• The number of times the wave repeats itself

during a unit of time is called the frequency• Units of measure are usually the number of

cycles (repetitions) per second• Cycles per second are called Hertz• frequency = 1 / period

Analog Signal

• Each signal has an amplitude or level

Analog Signal

• Each Signal has a frequency

Analog Signal

• Signal can also have a phase shift

Encoding bits with Analog Signals

• Encoding using amplitude change

• Encoding by changing frequency

• Encoding by changing signal phase

Encoding bits with Analog Signals

Bandwidth• Bandwidth is a measure of channel or circuit

capacity• Bandwidth is in part determined by the number

of times a signal changes value or frequency• Frequency is measured in Hertz• Both digital and Analog signals have frequency• Every circuit or channel has a frequency limit

Due to Hardware Due to limitation of media Artificially introduced

• Data bandwidth is the capacity of a circuit to carry data and is usually measure in bits per second or bps

Bandwidth

• Consider a regular home phone line This is an analog circuit The frequency of that signal is limited by the

phone company The circuit carries sound in frequencies

between 300 Hz and 3300 Hz For separation the phone company allocates

4000 Hz for each phone line or phone call

Baud Rate

• Speed of a circuit is measured in baud

• Baud The number of times a circuit can change

value If each change represents 1 bit then

• Baud Rate = Bit Rate

In all our previous examples• Baud Rate = Bit Rate

Suppose we could create a number of levels, for example, say we had 8 voltages levels we could use

000

001

010

011

100

101

110

111

Suppose we receive

011

000

100

111

000

001

110

010

011 000 100 111 000 001 110 010

Maximum Data Rateof a Channel

** Harry Nyquist (1924) showed

Max data rate (bps) = 2 * H * log2(V)

WhereH = frequency of channel in HertzV = number of distinct signal levels

Example – Voice grade line - 4000 Hz (3000 usable)If we use 2 signal levels

MDR = 2 * 3000 * log2(2) = 6,000 bps

Nyquist’s Result

• This result assumes a noiseless channel• We never have a ‘clean’ channel

Noise Distortion Attenuation Interference

• Circuits have a rating called

Signal to Noise Ratio – measured in decibels

S = measure of the power of the signalN = measure of the power of the noise

Limitations of Hardware

• Regardless of transmission type, signals degrade over distances Level or power of signal may degrade, called

attenuation Shape of signal may degrade, especially digital

signals. This is called distortion. Interference from external sources (NEXT)

• So, how fast can we send bits? How short can we make a ‘bit time’?

• Maximum bit rate is therefore limited by the media and the sensitivity of hardware to be able to accurately recreate transmitted bits

AttenuationSignals on any media are subject loss of signal strength

over distances

Let P0 = initial power of a signal Pm = measured power at some

distance of m unit

Power ratio =

Common measure is decibels

Pm

P0

Pm

PDB 010log*10***

Maximum Data Ratefor ‘noisy’ channel

Signal to Noise ratio = S/N Decibel = 10 * log10(S/N)

Claude Shannon (1948)

Maximum data rate of a channel ** = H * log2( 1 + S/N) H = frequency of channel

Example – voice grade line Signal to noise is approximately 30 DB

Mdr = 3000 * log2( 1 + 1000) = 3000 * log2( 1001 )

= 30,000 (approximately)

Regardless of the number of signals used

Modems

• Encode digital values over analog circuits• To encode data, modems use combinations of

Amplitude Shift Keying (ASK) Frequency Shift Keying (FSK) Phase Shift Keying (PSK)

• Typically, a carrier frequency is provided and various SHIFT KEYING is applied

• For half duplex we could have only one carrier• For full duplex, we need two carriers, one for

each direction

Modems

• There have been many modem standards over time

• Early modems used a form of frequency shifting – Ex 300 bps full duplex modem Originating modem

• Sine wave at 1070 Hz for a 0 bit• Sine wave at 1270 Hz for a 1 bit

Answering modem• Sine wave at 2025 Hz for a 0 bit• Sine wave at 2225 Hz for a 1 bit

Modems

• Later modems used combinations of shift keying by combining PSK and ASK

• Some common 9600 bps modems used 12 phase shifts at 1 amplitude 4 phase shift at a second amplitude Combination of 16 different states Called Quadrature Amplitude Modulation (QAM) Baud rate was 2400 cycles per second Each of the 16 states represented 4 bits

Modem Constellation

Modems

Standard Baud Bit Modulation rate rate Technique

V.21 300 300 FSKV.22 600 600/1200 PSKV.22bis 600 1200/2400 QAMV.23 1200 1200 FSKV.32 2400 4800/9600 QAM/TCMV.34 2400 28,800 V.90 2400 56,000V.92 2400 56,000

Modems

• To improve performance, compression and error correction standards developed

• Two compression standards in in vogue V.42bis MNP 5

• Two error correction standards V.42 MNP 4