Datornätverk A – lektion 4MKS B – lektion 4

•Kapitel 5: Modulation.

4.3 Sampling4.3 Sampling

Pulse Amplitude ModulationPulse Code ModulationSampling Rate: Nyquist TheoremHow Many Bits per Sample?Bit Rate

PCM = Pulse Code Modulation = Digitalisering av analoga signaler och seriell överföring

Sampler

AD-omvand-lare med

seriell utsignal

011011010001...

DA-omvandlare

Antiviknings-filter

Interpola-tionsfilter

Sifferexempel från PSTN = publika telefonnätet:

3400-4000Hz

filter

8000sampelsper sek

8 bit per sampeldvs 64000 bpsper tfnsamtal

28 = 256spänningsnivåer

0

1

Mikrofon Högtalare

Exempel

En 6 sekunder lång ljudinspelning digitaliseras. Hur stor är inspelningens informationsmängd?

a) 22000 sampels/sekund, 256 kvantiseringsnivåer.

b) 22000 sampels/sekund, 16 kvantiseringsnivåer.

c) 5500 sampels/sekund, 256 kvantiseringsnivåer.

22000sampels * 6 s * 8 bit = 1056000bit.

22000sampels * 6 s * 4 bit = 528000bit.

5500sampels * 6 s * 8 bit = 264000bit.

Samplingsteoremet

f < fs/2• Den högsta frekvens som kan samplas är halva samplingsfrekvensen.

• Om man samplar högre frekvens än fs/2 så byter signalen frekvens, dvs det uppstår vikningsdistorsion (aliasing).

• För att undvika vikningsdistorsion så har man ett anti-vikningsfilter innan samplingen, som tar bort frekvenser över halva samplingsfrekvensen.

• Interpolationsfiltret används vid rekonstruktion av den digitala signalen för att ”gissa” värden mellan samplen.

• Ett ideal interpolationsfilter skulle kunna återskapa den samplade signalen perfekt om den uppfyller samplingsteoremet. I verkligheten finns inga ideala filter.

• Följdregel: Nyqvist’s sats säger att max datahastighet = 2B2log M, där M är antal nivåer, och B är signalens bandbredd, oftast lika med signalens övre gränsfrekvens.

Figure 4.18 PAM

Pulse amplitude modulation has some Pulse amplitude modulation has some applications, but it is not used by itself applications, but it is not used by itself in data communication. However, it is in data communication. However, it is the first step in another very popular the first step in another very popular

conversion method called conversion method called pulse code modulation.pulse code modulation.

Note:Note:

Figure 4.19 Quantized PAM signal

Figure 4.20 Quantizing by using sign and magnitude

According to the Nyquist theorem, the According to the Nyquist theorem, the sampling rate must be at least 2 times sampling rate must be at least 2 times

the highest frequency.the highest frequency.

Note:Note:

Example 4Example 4

What sampling rate is needed for a signal with a bandwidth of 10,000 Hz (1000 to 11,000 Hz)?

SolutionSolution

The sampling rate must be twice the highest frequency in the signal:

Sampling rate = 2 x (11,000) = 22,000 samples/sSampling rate = 2 x (11,000) = 22,000 samples/s

Example 5Example 5

A signal is sampled. Each sample requires at least 12 levels of precision (+0 to +5 and -0 to -5). How many bits should be sent for each sample?

SolutionSolution

We need 4 bits; 1 bit for the sign and 3 bits for the value. A 3-bit value can represent 23 = 8 levels (000 to 111), which is more than what we need. A 2-bit value is not enough since 22 = 4. A 4-bit value is too much because 24 = 16.

Example 6Example 6

We want to digitize the human voice. What is the bit rate, assuming 8 bits per sample?

SolutionSolution

The human voice normally contains frequencies from 0 to 4000 Hz. Sampling rate = 4000 x 2 = 8000 samples/sSampling rate = 4000 x 2 = 8000 samples/s

Bit rate = sampling rate x number of bits per sample Bit rate = sampling rate x number of bits per sample = 8000 x 8 = 64,000 bps = 64 Kbps= 8000 x 8 = 64,000 bps = 64 Kbps

Distorsion till följd av digitalisering

• Vikningsdistorsion○ Inträffar om man inte filtrerar bort frekvenser som är högre än

halva samplingsfrekvensen.

• Kvantiseringsdistorsion (kvantiseringsbrus)○ Avrundningsfelet låter ofta som ett brus.

○ Varje extra bit upplösning ger dubbelt så många spänningsnivåer, vilket ger en minskning av kvantiseringsdistorsionen med 6 dB. 16 bit upplösning ger ett signal-brus-förhållande på ca 16*6 = 96 dB (beroende på hur man mäter detta förhållande.)

○ Svaga ljud avrundas bort, eller dränks i kvantiseringsbruset.

Informationsmängd

• N bit kan representera 2N alternativa värden eller koder.Ex: ASCII-kodens 7 bitar kan representera 27 = 2·2 ·2 ·2 ·2 ·2 ·2 = 128

tecken.

• En kod som kan anta M alternativa värden har informationsmängden

2 2log

log

log.M

M

Ex: ISO-latinkodens 256 tecken kräver 2log 256 = 8 bit per tecken.

Figure 5.25 Types of analog-to-analog modulation

Figure 5.26 Amplitude modulation

Figure 5.29 Frequency modulation

Modulation och demodulation

• Baudrate = antal symboler per sekund. Enhet: baud eller symboler/sekund.

• Bitrate = datahastighet. Enhet: bps eller bit/s.

• Vid många modulationsformer t.ex. s.k. ASK, PSK, och QAM är signalens bandbredd = symbolhastigheten.

• Vid FSK är bandbredden vanligen större.

Digitala modulationsmetoder

Binär signal

ASK = Amplitude Shift Keying (AM)

FSK = Frequency Shift Keying (FM)

PSK = Phase Shift Keying (PSK)

0 0.5 1-1

0

1

time [milliseconds]

00

0 0.5 1-1

0

1

time [milliseconds]

01

0 0.5 1-1

0

1

time [milliseconds]

11

0 0.5 1-1

0

1

time [milliseconds]

10

0 0.5 1 1.5 2 2.5 3 3.5 4-1

0

1

time [millisecond]

Exempel 1:Till höger visas fyra symboler som används av ett s.k. 4PSK-modem (PSK=Phase Shift Keying). De fyra symbolerna representerar bitföljderna 00, 01, 11 resp 10.

a) Nedan visas utsignalen från det sändande modemet. Vilket meddelande, dvs vilken bitsekvens, överförs?

b) Tidsaxeln är graderad i tusendels sekunder. Vad är symbolhastigheten i baud eller symboler/sekund?

c) Vad är bithastigheten i bit per sekund (bps)?Svar: 1/1ms = 1000 symber per sekund = 1kbaud.

Svar: 2000bps.

Svar: 11 00 10 10.

0 0.005 0.01-2

0

2000

0 0.005 0.01-2

0

2001

0 0.005 0.01-2

0

2011

0 0.005 0.01-2

0

2010

0 0.005 0.01-2

0

2100

0 0.005 0.01-2

0

2101

0 0.005 0.01-2

0

2111

0 0.005 0.01-2

0

2110

Exempel 2:Nedan visas åtta symboler som används av ett s.k. 8QAM-modem (QAM=Quadrature Amplitude Modulation). Symbolerna i övre raden representerar bitföljderna 000, 001, 011 resp 010 (från vänster till höger). Undre raden representerar 100, 101, 111 resp 110.

a) Nedan visas utsignalen från det sändande modemet. Vilketmeddelande, dvs vilken bitsekvens, överförs?

b) Tidsaxlarna är graderad i sekunder. Vad är symbolhastigheten i baudeller symboler/sekund?

c) Vad är bithastigheten i bit per sekund (bps)?

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-2

0

2

Tid [sekunder]

Spä

nning [V

olt]

Modulatorns utsignal

Forts exempel 2:

Example 1Example 1

An analog signal carries 4 bits in each signal unit. If 1000 signal units are sent per second, find the baud rate and the bit rate

SolutionSolution

Baud rate = 1000 bauds per second (baud/s)Baud rate = 1000 bauds per second (baud/s)Bit rate = 1000 x 4 = 4000 bpsBit rate = 1000 x 4 = 4000 bps

Figure 5.4 Relationship between baud rate and bandwidth in ASK

Figure 5.3 ASK

Example 3Example 3

Find the minimum bandwidth for an ASK signal transmitting at 2000 bps. The transmission mode is half-duplex.

SolutionSolution

In ASK the baud rate and bit rate are the same. The baud rate is therefore 2000. An ASK signal requires a minimum bandwidth equal to its baud rate. Therefore, the minimum bandwidth is 2000 Hz.

Example 4Example 4

Given a bandwidth of 5000 Hz for an ASK signal, what are the baud rate and bit rate?

SolutionSolution

In ASK the baud rate is the same as the bandwidth, which means the baud rate is 5000. But because the baud rate and the bit rate are also the same for ASK, the bit rate is 5000 bps.

Example 6Example 6

Find the minimum bandwidth for an FSK signal transmitting at 2000 bps. Transmission is in half-duplex mode, and the carriers are separated by 3000 Hz.

SolutionSolution

For FSK BW = baud rate + fBW = baud rate + fc1c1 f fc0c0

BW = bit rate + fc1 BW = bit rate + fc1 fc0 = 2000 + 3000 = 5000 Hz fc0 = 2000 + 3000 = 5000 Hz

Example 7Example 7

Find the maximum bit rates for an FSK signal if the bandwidth of the medium is 12,000 Hz and the difference between the two carriers is 2000 Hz. Transmission is in full-duplex mode.

SolutionSolution

Because the transmission is full duplex, only 6000 Hz is allocated for each direction. BW = baud rate + fc1 BW = baud rate + fc1 fc0 fc0 Baud rate = BW Baud rate = BW (fc1 (fc1 fc0 ) = 6000 fc0 ) = 6000 2000 = 4000 2000 = 4000But because the baud rate is the same as the bit rate, the bit rate is 4000 bps.

Figure 5.8 PSK

Figure 5.5 Solution to Example 5

Figure 5.6 FSK

Figure 5.9 PSK constellation

Figure 5.10 The 4-PSK method

Figure 5.11 The 4-PSK characteristics

Figure 5.12 The 8-PSK characteristics

Figure 5.13 Relationship between baud rate and bandwidth in PSK

Example 9Example 9

Given a bandwidth of 5000 Hz for an 8-PSK signal, what are the baud rate and bit rate?

SolutionSolution

For PSK the baud rate is the same as the bandwidth, which means the baud rate is 5000. But in 8-PSK the bit rate is 3 times the baud rate, so the bit rate is 15,000 bps.

Quadrature amplitude modulation is a combination of ASK and PSK so that a

maximum contrast between each signal unit (bit, dibit, tribit, and so on)

is achieved.

Note:Note:

Figure 5.14 The 4-QAM and 8-QAM constellations

Figure 5.15 Time domain for an 8-QAM signal

Figure 5.16 16-QAM constellations

Figure 5.17 Bit and baud

Table 5.1 Bit and baud rate comparison

ModulationModulation UnitsUnits Bits/BaudBits/Baud Baud rateBaud rate Bit Rate

ASK, FSK, 2-PSKASK, FSK, 2-PSK Bit 1 N N

4-PSK, 4-QAM4-PSK, 4-QAM Dibit 2 N 2N

8-PSK, 8-QAM8-PSK, 8-QAM Tribit 3 N 3N

16-QAM16-QAM Quadbit 4 N 4N

32-QAM32-QAM Pentabit 5 N 5N

64-QAM64-QAM Hexabit 6 N 6N

128-QAM128-QAM Septabit 7 N 7N

256-QAM256-QAM Octabit 8 N 8N

Example 10Example 10

A constellation diagram consists of eight equally spaced points on a circle. If the bit rate is 4800 bps, what is the baud rate?

SolutionSolution

The constellation indicates 8-PSK with the points 45 degrees apart. Since 23 = 8, 3 bits are transmitted with each signal unit. Therefore, the baud rate is 4800 / 3 = 1600 baud

Example 11Example 11

Compute the bit rate for a 1000-baud 16-QAM signal.

SolutionSolution

A 16-QAM signal has 4 bits per signal unit since log216 = 4. Thus,

1000·4 = 4000 bps

Example 12Example 12

Compute the baud rate for a 72,000-bps 64-QAM signal.

SolutionSolution

A 64-QAM signal has 6 bits per signal unit since log2 64 = 6. Thus, 72000 / 6 = 12,000 baud

5.2 Telephone Modems

Modem Standards

A telephone line has a bandwidth of almost 2400 Hz for data transmission.

Note:Note:

Figure 5.18 Telephone line bandwidth

Modem stands for modulator/demodulator.

Note:Note:

Figure 5.19 Modulation/demodulation

Figure 5.20 The V.32 constellation and bandwidth

Figure 5.21 The V.32bis constellation and bandwidth

Figure 5.22 Traditional modems

Figure 5.23 56K modems

5.3 Modulation of Analog Signals5.3 Modulation of Analog Signals

Amplitude Modulation (AM)

Frequency Modulation (FM)

Phase Modulation (PM)

Figure 5.24 Analog-to-analog modulation

The total bandwidth required for AM can be determined from the bandwidth

of the audio signal: BWt = 2 x BWm.

Note:Note:

Example 7Example 7

Consider a noiseless channel with a bandwidth of 3000 Hz transmitting a signal with two signal levels. The maximum bit rate can be calculated as

BitBit Rate = 2 Rate = 2 3000 3000 log log22 2 = 6000 bps 2 = 6000 bps

Shannons regel

Kanalkapaciteten C är max antal bit per sekund vid bästa möjliga modulationsteknik och felrättande kodning:

C = B log2 (1+S/N),

där B är ledningens bandbredd i Hertz (oftast ungefär lika med övre gränsfrekvensen), S är nyttosignalens medeleffekt i Watt och N (noice) är bruseffekten i Watt.

Example 8Example 8

Consider the same noiseless channel, transmitting a signal with four signal levels (for each level, we send two bits). The maximum bit rate can be calculated as:

Bit Rate = 2 x 3000 x logBit Rate = 2 x 3000 x log22 4 = 12,000 bps 4 = 12,000 bps

Example 9Example 9

Consider an extremely noisy channel in which the value of the signal-to-noise ratio is almost zero. In other words, the noise is so strong that the signal is faint. For this channel the capacity is calculated as

C = B logC = B log22 (1 + SNR) = B log (1 + SNR) = B log22 (1 + 0) (1 + 0)

= B log= B log22 (1) = B (1) = B 0 = 0 0 = 0

Example 10Example 10

We can calculate the theoretical highest bit rate of a regular telephone line. A telephone line normally has a bandwidth of 3000 Hz (300 Hz to 3300 Hz). The signal-to-noise ratio is usually 3162. For this channel the capacity is calculated as

C = B logC = B log22 (1 + SNR) = 3000 log (1 + SNR) = 3000 log22 (1 + 3162) (1 + 3162)

= 3000 log= 3000 log22 (3163) (3163)

C = 3000 C = 3000 11.62 = 34,860 bps 11.62 = 34,860 bps

Example 11Example 11

We have a channel with a 1 MHz bandwidth. The SNR for this channel is 63; what is the appropriate bit rate and signal level?

SolutionSolution

C = B logC = B log22 (1 + SNR) = 10 (1 + SNR) = 1066 log log22 (1 + 63) = 10 (1 + 63) = 1066 log log22 (64) = 6 Mbps (64) = 6 Mbps

Then we use the Nyquist formula to find the number of signal levels.

4 Mbps = 2 4 Mbps = 2 1 MHz 1 MHz log log22 LL L = 4 L = 4

First, we use the Shannon formula to find our upper First, we use the Shannon formula to find our upper limit.limit.