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### Digital transmission fundamentals (chap.3 & 4)

- What is digital transmission?
- Why digital transmission?
- How to represent different information
- in digital format?
- What is Bit rate/bandwidth?
- What are the properties of different media?
- How to do error detection and correction?
- Multiplexing?

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What is digital transmission?

- Analog transmission
- Continuous waveform
- Digital representation and transmission
- Discrete binary sequence/pulses
- 1: a rectangular pulse of amplitude 1 and of duration 0.125 milliseconds
- 0: a rectangular pulse of amplitude -1 and of duration of 0.125 milliseconds.

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Figure 3.13

Analog Digital ? --PCM

- Analog signal such as voice/music:
- continuous waveform, i.e, variations in air pressure.
- Bandwidth: a measure of how fast the signal varies, i.e., cycles/second, or Hertz.
- Two stages:
- Sampling: for bandwidth W, minimum sampling rate is 2W.
- Quantizing: how many levels to represent a sample.
- Example: W=4 kHz, sampling rate=8K samples/second. Sample period is T=1/8000=125 microseconds. Suppose 8 bits/sample (256 levels), then PCM bit rate is 8000*8=64 kbps.

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Why digital?

- For analog:
- output should reproduce the input exactly. No distortion.
- Repeater amplifies noise, difficult task.
- Too much repeaters may make noise too large, so limited distance
- Cost is high.
- Basic voice/telephone service.
- For digital:
- Not exact, as long as can distinguish 1 or o.
- Digital regenerator generates pure digital numbers, easy.
- No limitation on digital regenerators, no distance limitation.
- Cost is low.
- More other services, easily multiplexing, more functions.

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Sent

(a) Analog transmission: all details must be reproduced accurately

- e.g. AM, FM, TV transmission

(b) Digital transmission: only discrete levels need to be reproduced

Received

Sent

- e.g digital telephone, CD Audio

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Figure 3.6

+

residual noise

Attenuated & distorted signal

+

noise

Amp.

Equalizer

Repeater

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Figure 3.8

& Signal

Regenerator

Amplifier

Equalizer

Timing

Recovery

Distorted Digital signal is easy to restore by regenerator.

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Figure 3.9

Digital representations for different information

- Text: ASCII
- Scanned WB documents:
- A4 paper, 200 X 100 pixels/inch. 256KB.
- Color pictures/images:
- 8 X 10 inch photo, 400 X400 pixels/inch. 38.4MB.
- Voice: PCM/ADPCM, 4kHz, 64kbps (this as well the followings called stream)
- Music/Audio:
- MPEG/MP3, 16-24 kHz, 512-748 kbps.
- Video: a sequence of pictures (moving pictures)
- H.261, 176 X 144 pixels/frame, 10-30 frames/second, 2 Mbps.
- MPEG-2, 720 X 480 pixels/frame, 30 frames/second, 249 Mbps.
- Compression:
- 249Mbps 2 – 6 Mbps.
- Compression cost, but reduce the transmission cost.

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Receiver

Communication channel

(Digital) Transmission System

Transmitter

Receiver

0110101…

0110101…

Communication channel

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Figure 3.5

Basic properties of digital transmission systems

- Bit rate or transmission speed R: bits/second.
- Can be viewed as cross-section of the channel: the higher R is, the larger the volume of the channel.
- Bandwidth of a signal: Ws
- The range of frequencies contained in the signal.
- Bandwidth of channel: Wc
- The range of input frequencies passed by the channel.
- Wc limits Ws that can pass through the channel.

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Basic properties of digital transmission systems (cont.)

- Theory: if Wc=W, then the narrowest pulse has duration =1/2W. Thus, the maximum rate for pulses is rmax=2W pulses/second.
- If transmitting binary information by sending two kinds of pulses: +A for 1 and _A for 0, then the system bite rate is R=2W pulses/second * 1bit/pluse =2W bits/second.
- If pulses can be multiple levels (M=4): -A, -A/3, +A/3, +A for (00, 01, 10,11), then each pulse can represent 2 bits. So R=4W bps.
- If multiple levels M=2m, then R=2W *m=2Wm bps.
- Theoretically, the bit rate R can be increased without limits as long as we increase M.

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Shannon Channel Capacity

- Unfortunately, in practice, R is greatly limited:
- Levels can not be measured accurately if too many levels.
- there exists noise in real world.
- Signal –to-noise ratio SNR= average signal power/average noise power.
- SNR(dB)=10log10SNR (decibels).
- So Shannon Channel Capacity:
- C (i.e., maximum reliable bite rate R )
- = W log 2(1+SNR) bits/second.

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Signal

Noise

High

SNR

t

t

t

Noise

Signal + noise

Signal

Low

SNR

t

t

t

Average signal power

SNR =

Average noise power

SNR (dB) = 10 log10 SNR

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Figure 3.11

Four signal levels

Eight signal levels

When too many levels: 1. difficult to measure

2. noise will easily affect the value.

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Figure 3.32

Telephone Modem example

- W=3.4 kHz, SNR=10,000, SNR(dB)=40 dB.
- C=3400log2(1+10000) =45200 bits/second.
- So telephone channel is at 45.2kbps.
- Interesting point: V.90 model’s rate: 56kbps.
- Inbound to network: 33.6kbps
- Analog to digital, quantization noise, SNR=39 dB.
- Outbound to user from ISP: already in digital.

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Line Coding

- How to converting binary information sequence into digital signal.
- Consideration of choices: (apart from bit rate)
- Average transmission power
- Ease of bit timing (synchronization)
- Prevention of dc and low-frequency content
- Ability of error-detection
- Immunity to noise and interference
- Cost and complexity.

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Various line coding

- Unipolar nonreturn-to-zero encoding (NRZ)
- 1: +A, 0: 0 voltage.
- Polar NRZ:
- 1: +A/2, 0: -A/2
- Bipolar encoding:
- 0: 0 voltage, consecutive 1s are alternately mapped to +A/2, -A/2.
- NRZ inverted (Differential encoding)
- 1: a transition at the beginning of a bit time.
- 0: no transition.
- Manchester encoding (used in Ethernet):
- 1: a transition from +A/2 to –A/2 in the middle of a bit time
- 0: a transition from -A/2 to +A/2 in the middle of a bit time
- Differential Manchester encoding (used in Token-ring networks):
- A transition in middle of each bit time
- 1: absence of transition
- 0: a transition at the beginning of an interval.

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1

0

1

1

1

1

0

0

Unipolar

NRZ

1: +A, 0: 0 voltage

Polar NRZ

1: +A/2, 0: -A/2

NRZ-Inverted

(Differential

Encoding)

1: transition, 0: not

Bipolar

Encoding

- 0: 0 voltage, 1s are alternately mapped to +A/2, -A/2.

Manchester

Encoding

1: a transition from +A/2 to –A/2, 0: otherwise

Differential

Manchester

Encoding

2 pulses/bit, 1 10, 001

- A transition in middle of each bit time, 1: absence of transition,0: a transition at the beginning.

Figure 3.35

mBnB encoding (n>m)

- Means “m bits information are mapped to n encodedbits.
- Manchester encoding is 1B2B.
- Optical transmission 4B5B.
- FDDI: 8B10B is used.

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Transmission Media

- Twisted Pair
- DSL, LAN (Ethernet), ISDN
- Coaxial Cable
- Cable TV, Cable Modem, Ethernet.
- Optical Fiber
- Backbone, LAN
- Radio Transmission
- Cellular network, Wireless LAN, Satellite network.
- Infrared Light
- IrDA Links

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Transmission Delay

- L number of bits in message
- R bps speed of digital transmission system
- L/R time to transmit the information
- tprop time for signal to propagate across medium
- d distance in meters
- c speed of light (3x108 m/s in vacuum)

Use data compression to reduce L

Use higher speed modem to increase R

Place server closer to reduce d

Delay = tprop + L/R = d/c + L/R seconds

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Error Detection and Correction

- Error detection and retransmission
- When return channel is available
- Used in Internet
- Waste bandwidth
- Forward error correction (FEC)
- When the return channel is not available
- When retransmission incurs more cost
- Used in satellite and deep-space communication, as well as audio CD recoding.
- Require redundancy and processing time.

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Odd error detection using parity bit

- Seven data bit plus 1 parity bit
- 1011010 0
- 1010001 1
- Then any odd errors can be detected.

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Two-dimension parity checks

1 0 0 1 0 0

0 1 0 0 0 1

1 0 0 1 0 0

1 1 0 1 1 0

1 0 0 1 1 1

- Several information rows
- Last column: check bits for rows
- Last row: check bits for columns

Can detect one, two, three errors,

But not all four errors.

1 0 0 1 0 0

0 0 0 0 0 1

1 0 0 1 0 0

1 1 0 1 1 0

1 0 0 1 1 1

1 0 0 1 0 0

0 0 0 0 0 1

1 0 0 1 0 0

1 0 0 1 1 0

1 0 0 1 1 1

1 0 0 1 0 0

0 0 0 1 0 1

1 0 0 1 0 0

1 0 0 1 1 0

1 0 0 1 1 1

1 0 0 1 0 0

0 0 0 1 0 1

1 0 0 1 0 0

1 0 0 0 1 0

1 0 0 1 1 1

2 errors

3 errors

4 errors

1 error

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Internet Checksum

- IP packet, a checksum is calculated for the headers and put in a field in the header.
- Goal is easy/efficient to implement, make routers simple and efficient.
- Suppose m 16-bit words, b0, b1, …, bm-1
- Compute x= b0+ b1+ …+ bm-1 mod 216 -1
- Set checksum bm=-x
- Insert x in the checksum field
- Verify b0+ b1+ …+ bm-1 + bm=0 mod 216 -1.

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CRC (Cyclic Redundancy Check) (chapter 3.9.4)

- Based on polynomial codes and easily implemented using shift-register circuit
- Information bits, codewords, error vector are represented as polynomials with binary coefficients. On the contrary, the coefficients of a polynomial will be a binary string.
- Polynomial arithmetic is done modulo 2 with addition and subtraction being Exclusive-OR, so addition and subtraction is the same.
- Examples: 10110 x4 + x2 + x

01011 x3 + x + 1

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11000001 + 01100000 = 10100001

Multiplication:

00000011 * 00000111 = 00001001

1110

= q(x) quotient

x3 + x2 + x

00001011

) 01100000

Division:

x3 + x+ 1 ) x6 + x5

1011

x6 + x4 + x3

dividend

1110

divisor

1011

x5 + x4 + x3

1010

x5 + x3 + x2

1011

3

10

35 ) 122

x4 + x2

105

x4 + x2 + x

17

x

= r(x) remainder

polynomial arithmetic

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Figure 3.55

How to compute CRC

- There is a generator polynomial, g(x) of degree r, which the sender and receiver agree upon in advance.
- Suppose the information transmitted has m bits, i.e. i(x), then sender appends rzero at the end of information, i.e, xr i(x)
- Perform xr i(x) / g(x) to get remainder r(x) (and quotient q(x))
- Append the r bit string of r(x) to the end of m bit information to get m+r bit string, i.e., b(x), for transmission.
- b(x) = xr i(x)+ r(x) ( =g(x)q(x)+r(x)+r(x)=g(x)q(x) )
- When receiver receives the bit string, i.e., b’(x), it will divide b’(x)by g(x) , if the remainder is not zero, then error occurs.

--supposeno error occurs, then b’(x) = b(x), so b’(x)/g(x)

=b(x)/g(x)=g(x)q(x)/g(x)=q(x), remainder is 0.

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x3 + x2 + x

1110

Generator polynomial: g(x)= x3 + x + 1 1011

Information: (1,1,0,0) i(x) = x3 + x2

Encoding: x3i(x) = x6 + x5

x3 + x+ 1 x6 + x5

1011 ) 1100000

x6 + x4 + x3

1011

x5 + x4 + x3

1110

1011

x5 + x3 + x2

1010

x4 + x2

1011

x4 + x2 + x

x

010

- Transmitted codeword:
- b(x) = x6 + x5 + x
- b= (1,1,0,0,0,1,0)

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Figure 3.57

Example of CRC encoding (cont.)

If 1100010 is received, then 1100010 is divided by 1011,

The remainder will be zero (please verify yourself), so no error.

Suppose 1101010 is received, then let us do the division as follows:

1111

1011

) 1101010

1011

1100

1011

1111

1011

1000

1011

The remainder is not zero, so

error occurs.

11

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Typical standard CRC polynomials

- CRC-8: x8 + x2 + x + 1 ATM header error check
- CRC-16: x16+x12+x5+1 HDLC, XMODEM, V.41
- CRC-32: x32+x26+x23+x22+ IEEE 802, DoD, V.41,

x16+x12+x11+x10+ AAL5

x8+x7+x5+x4+x2+x+1

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Analysis of error detection power

Received poly: R(x)= b(x) +e(x), where e(x) is error poly.

1. Single errors: e(x) = xi0 in-1

If g(x) has more than one term, it cannot divide e(x)

2. Double errors:e(x) = xi+ xj0 i < jn-1

= xi(1 + xj-i)

If g(x) is primitive, it will not divide (1 + xj-i) for j-i 2n-k1

3. Odd number of errors:e(1) =1.

If g(x) has (x+1) as a factor, then g(1) = 0 and all codewords have an even number of 1s.

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Figure 3.60

The error detection capabilities of CRCs

- As long as the g(x) is selected appropriately
- All single errors
- All double errors
- All odd number of errors
- Burst error of length L, the probability that this burst error is undetectable = 1/2(L-2)
- CRC can be easily implemented in hardware
- One word about error correction: more powerful, but need more extra check bits, more slow, so not use as much as error detection.

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Statistical multiplexing

- Burst feature of user interactions makes dedicated communication lines inefficient
- Multiplexing multiple lines into one line which has generally more bandwidth.
- In order for coordinating the usage of the shared line, buffers are needed.
- Multiplexers are introduced for multiplexing

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Time-Division Multiplexing (TDM)

(a)

Dedicated Lines

A1

A2

B1

B2

C1

C2

(b)

Shared Line

B2

C2

A2

A1

C1

B1

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Figure 5.43

1

2

MUX

MUX

2

. . .

. . .

22

23

24

1

b

24

2

b

. . .

frame

24

24

T-1 carrier system uses TDM to carry 24 digital signals in telephone system

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Figure 4.4

1

2

2

n

n

Wavelength-division multiplexing (WDM)

1, 2, n

MUX

deMUX

Optical fiber

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Figure 4.1

Frequency-Division Multiplexing (FDM)

A

f

W

0

B

f

0

W

C

B

A

f

C

f

0

W

(a) Individual signals occupy W Hz

(b) Combined signal fits into channel bandwidth

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Figure 4.2

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