Digital Transmission

Fundamentals
Digital Representation of Information
Why Digital Communications?
Digital Representation of Analog Signals
Characterization of Communication Channels
Fundamental Limits in Digital Transmission
7-Layer OSI Reference Model
Application Application
End-to-End Protocols
Application Application
Layer Layer
Presentation Presentation
Layer Layer
Session Session
Layer Layer
Transport Transport
Layer Layer
Network Network Network Network
Layer Layer Layer Layer
Data Link Data Link Data Link Data Link
Layer Layer Layer Layer
Physical Physical Physical Physical
Layer Layer Layer Layer

Communicating End Systems

Digital Networks
l Digital transmission enables networks to
support many services
E-mail
TV

Telephone
Digital Transmission
Fundamentals
Digital Representation of
Information
Bits, numbers, information
l Bit: number with value 0 or 1
l n bits: digital representation for 0, 1, … , 2n
l Byte or Octet, n = 8
l Computer word, n = 16, 32, or 64
l n bits allows enumeration of 2n possibilities
l n-bit field in a header
l n-bit representation of a voice sample
l Message consisting of n bits
l The number of bits required to represent a message
is a measure of its information content
l More bits → More content
Block vs. Stream Information
Block Stream
l Information that occurs l Information that is
in a single block produced & transmitted
l Text message continuously
l Data file l Real-time voice
l JPEG image l Streaming video
l MPEG file
l Size = Bits / block l Bit rate = bits / second
or bytes/block l 1 kbps = 103 bps
l 1 kbyte = 210 bytes l 1 Mbps = 106 bps
l 1 Mbyte = 220 bytes l 1 Gbps =109 bps
l 1 Gbyte = 230 bytes
Stream Information
l A real-time voice signal must be digitized &
transmitted as it is produced
l Analog signal level varies continuously in time

Th e s p ee ch s i g n al l e v el v a r ie s w i th t i m(e)
Digitization of Analog Signal
l Sample analog signal in time and amplitude
l Find closest approximation
Original signal
Sample value

7D/2 Approximation
5D/2
3 bits / sample

3D/2
D/2
-D/2
-3D/2
-5D/2
-7D/2

Rs = Bit rate = # bits/sample x # samples/second

 Video Signal
l Sequence of picture frames
l Each picture digitized &
compressed
l Frame repetition rate
l 10-30-60 frames/second
depending on quality
l Frame resolution
l Small frames for 30 fps
videoconferencing
l Standard frames for
conventional broadcast TV
l HDTV frames

Rate = M bits/pixel x (WxH) pixels/frame x F frames/second

Video Frames
176
QCIF videoconferencing 144 at 30 frames/sec =
760,000 pixels/sec

720
Broadcast TV at 30 frames/sec =
480
10.4 x 106 pixels/sec

1920
HDTV at 30 frames/sec =

1080 67 x 106 pixels/sec

Digital Video Signals
Type Method Format Original Compressed

Video H.261176x144 or 2-36 64-1544

Confer- 352x288 pix Mbps kbps
ence @10-30
fr/sec
Full MPEG 720x480 pix 249 2-6 Mbps
Motion 2 @30 fr/sec Mbps

HDTV MPEG 1920x1080 1.6 19-38 Mbps

2 @30 fr/sec Gbps
Transmission of Stream
Information
l Constant bit-rate
l Signals such as digitized telephone voice produce
a steady stream: e.g. 64 kbps
l Network must support steady transfer of signal,
e.g. 64 kbps circuit
l Variable bit-rate
l Signals such as digitized video produce a stream
that varies in bit rate, e.g. according to motion and
detail in a scene
l Network must support variable transfer rate of
signal, e.g. packet switching or rate-smoothing
with constant bit-rate circuit
 Communication
Networks and Services
Why Digital Communications?
A Transmission System
Transmitter Receiver

Communication channel

Transmitter
l Converts information into signal suitable for transmission
l Injects energy into communications medium or channel
l Telephone converts voice into electric current
l Modem converts bits into tones
Receiver
l Receives energy from medium
l Converts received signal into form suitable for delivery to user
l Telephone converts current into voice
l Modem converts tones into bits
Transmission Impairments
Transmitted Received
Transmitter Signal Signal Receiver

Communication channel

Communication Channel Transmission Impairments

l Pair of copper wires l Signal attenuation
l Coaxial cable l Signal distortion

l Radio l Spurious noise

l Light in optical fiber l Interference from other
l Light in air signals
l Infrared
 Analog Long-Distance
Communications
Transmission segment

Source Repeater ... Repeater Destination

l Each repeater attempts to restore analog signal to

its original form
l Restoration is imperfect
l Distortion is not completely eliminated
l Noise & interference is only partially removed
l Signal quality decreases with # of repeaters
l Communications is distance-limited
l Analogy: Copy a song using a cassette recorder
Analog vs. Digital Transmission
Analog transmission: all details must be reproduced accurately
Distortion
Sent Attenuation Received

Digital transmission: only discrete levels need to be reproduced

Sent Distortion Received

Simple Receiver:
Attenuation
Was original pulse
positive or
negative?
 Digital Long-Distance
Communications
Transmission segment

Source Regenerator ... Regenerator Destination

l Regenerator recovers original data sequence and

retransmits on next segment
l Can design so error probability is very small
l Then each regeneration is like the first time!
l Analogy: copy an MP3 file
l Communications is possible over very long distances
l Digital systems vs. analog systems
l Less power, longer distances, lower system cost
l Monitoring, multiplexing, coding, encryption, protocols…
Digital Transmission
Fundamentals
Digital Representation of
Analog Signals
Digitization of Analog Signals
1. Sampling: obtain samples of x(t) at uniformly
spaced time intervals
2. Quantization: map each sample into an
approximation value of finite precision
l Pulse Code Modulation: telephone speech
l CD audio
3. Compression: to lower bit rate further, apply
additional compression method
l Differential coding: cellular telephone speech
l Subband coding: MP3 audio
 Sampling Rate and Bandwidth
l A signal that varies faster needs to be sampled
more frequently
l Bandwidth measures how fast a signal varies
10 10 1 0 1 0 11 1 1 0 000
x1(t) x2(t)
... ... ... ...

t t

1 ms 1 ms

l What is the bandwidth of a signal?

l How is bandwidth related to sampling rate?
Periodic Signals
l A periodic signal with period T can be represented
as sum of sinusoids using Fourier Series:
x(t) = a0 + a1cos(2pf0t + f1) + a2cos(2p2f0t + f2) + …
+ akcos(2pkf0t + fk) + …

“DC” fundamental
long-term frequency f0=1/T kth harmonic
average first harmonic

•|ak| determines amount of power in kth harmonic

•Amplitude specturm |a0|, |a1|, |a2|, …

 Example Fourier Series
10 10 1 0 1 0 11 1 1 0 000
x1(t) x2(t)
... ... ... ...

t t

T2 =0.25 ms T1 = 1 ms

4 4
x1(t) = 0 + cos(2p4000t) x2(t) = 0 + cos(2p1000t)
p p
4 4
+ cos(2p3(4000)t) + cos(2p3(1000)t)
3p 3p
4 4
+ cos(2p5(4000)t) + … + cos(2p5(1000)t) + …
5p 5p

Only odd harmonics have power

Spectra & Bandwidth
Spectrum of x1(t)
l Spectrum of a signal: 1.2

magnitude of amplitudes as 1

0.8

a function of frequency 0.6

0.4

l x1(t) varies faster in time & 0.2

has more high frequency

0

9
12

15

18

21

24

27

30

33

36

39

42
content than x2(t) frequency (kHz)

l Bandwidth Ws is defined as Spectrum of x2(t)

range of frequencies where 1.2

1
a signal has non-negligible 0.8

power, e.g. range of band 0.6

0.4

that contains 99% of total 0.2

signal power
0
0

9
12

15

18

21

24

27

30

33

36

39

42
frequency (kHz)
Bandwidth of General Signals
“speech”

s (noisy ) |p (air stopped) | ee (periodic) | t (stopped) | sh (noisy)

l Not all signals are periodic

X(f)
l E.g. voice signals varies
according to sound
l Vowels are periodic, “s” is
noiselike
l Spectrum of long-term signal
l Averages over many sounds,
many speakers f
l Involves Fourier transform 0 Ws
l Telephone speech: 4 kHz
l CD Audio: 22 kHz
Sampling Theorem

Nyquist: Perfect reconstruction if sampling rate 1/T > 2Ws

(a) x(t) x(nT)

t Sampler t

(b)

x(nT) x(t)

t Interpolation t
filter
Digital Transmission of Analog
Information
2W samples / sec m bits / sample
Analog Sampling Quantization
source (A/D)

Original x(t) 2W m bits/sec

Bandwidth W

Transmission
or storage
Approximation y(t)

Display Interpolation Pulse

or filter
playout generator

2W samples / sec
Quantization of Analog Samples
3.5D
Quantizer maps input
output y(nT) into closest of 2m
2.5D
1.5D representation values
0.5D
-4D -3D -2D -D
-0.5D 3D 4D
D 2D
input x(nT)
Quantization error:
-1.5D
-2.5D “noise” = x(nT) – y(nT)
-3.5D

Original signal
Sample value
7D/2 Approximation
3 bits / sample

5D/2
3D/2
D/2
-D/2
-3D/2
-5D/2
-7D/2
Example: Voice & Audio
Telephone voice CD Audio
l Ws = 4 kHz → 8000 l Ws = 22 kHertz → 44000
samples/sec samples/sec
l 8 bits/sample l 16 bits/sample

l Rs=8 x 8000 = 64 kbps l Rs=16 x 44000= 704 kbps

per audio channel
l Cellular phones use l MP3 uses more powerful

more powerful compression algorithms:

compression 50 kbps per audio
algorithms: 8-12 kbps channel
Quantizer Performance
M = 2m levels, Dynamic range( -V, V) Δ = 2V/M
error = x(nT)-y(nT)=e(nT)

2
... -2D D 3D ... input
D 2D

V x(nT)
-V
2

If the number of levels M is large, then the error is

approximately uniformly distributed between (-Δ/2, Δ/2)
Average Noise Power = Mean Square Error:
Δ
1 Δ2
∫
2
σe =2 x2 dx =
Δ Δ 12
2
Quantizer Performance
Figure of Merit:
Signal-to-Noise Ratio = Avg signal power / Avg noise power
Let sx2 be the signal power, then
sx2 12 s 2 sx sx
SNR = = x
= 3( )2 M2
D2/12 4V2/M2 = 3( )2 22m
V V
The ratio V/sx » 4

The SNR is usually stated in decibels:

SNR dB = 10 log10 sx2/se2 = 6m + 10 log10
3sx2/V2
SNR dB = 6m - 7.27 dB for V/sx = 4.
Example: Telephone Speech
W = 4KHz, so Nyquist sampling theorem
Þ 2W = 8000 samples/second
Suppose error requirement = 1% error
SNR = 10 log(1/.01)2 = 40 dB
Assume V/sx =4, then
40 dB = 6m – 7
Þ m = 8 bits/sample

PCM (“Pulse Code Modulation”) Telephone

Speech:
Bit rate= 8000 x 8 bits/sec= 64 kbps
Digital Transmission
Fundamentals
Characterization of
Communication Channels
Communications Channels
l A physical medium is an inherent part of a
communications system
l Copper wires, radio medium, or optical fiber
l Communications system includes electronic or
optical devices that are part of the path followed by
a signal
l Equalizers, amplifiers, signal conditioners
l By communication channel we refer to the combined
end-to-end physical medium and attached devices
l Sometimes we use the term filter to refer to a
channel especially in the context of a specific
mathematical model for the channel
Communications Channel
Transmitter Transmitted Received
Signal Signal Receiver

Communication channel

Signal Bandwidth Transmission Impairments

l In order to transfer data l Signal attenuation
faster, a signal has to vary l Signal distortion
more quickly.
l Spurious noise
Channel Bandwidth
l Interference from other
l A channel or medium has signals
an inherent limit on how fast
l Limits accuracy of
the signals it passes can
vary measurements on received
signal
l Limits how tightly input
pulses can be packed
Frequency Domain Channel
Characterization

x(t)= Aincos 2pft y(t)=Aoutcos (2pft + j(f))

Channel
t t
Aout
A(f) = Ain

l Apply sinusoidal input at frequency f

l Output is sinusoid at same frequency, but attenuated & phase-shifted
l Measure amplitude of output sinusoid (of same frequency f)
l Calculate amplitude response
l A(f) = ratio of output amplitude to input amplitude
l If A(f) ≈ 1, then input signal passes readily
l If A(f) ≈ 0, then input signal is blocked
l Bandwidth Wc is range of frequencies passed by channel
Ideal Low-Pass Filter
l Ideal filter: all sinusoids with frequency f<Wc are
passed without attenuation and delayed by t seconds;
sinusoids at other frequencies are blocked

y(t)=Aincos (2pft - 2pft )= Aincos (2pf(t - t )) = x(t-t)

Amplitude Response Phase Response

1 j(f) = -2pft
1/ 2p
0
f

Wc f
Time-domain Characterization
h(t)

Channel
t
0 t
td

l Time-domain characterization of a channel requires

finding the impulse response h(t)
l Apply a very narrow pulse to a channel and observe
the channel output
l h(t) typically a delayed pulse with ringing
l Interested in system designs with h(t) that can be
packed closely without interfering with each other
 Nyquist Pulse with Zero
Intersymbol Interference
l For channel with ideal low pass amplitude response of
bandwidth Wc, the impulse response is a Nyquist pulse
h(t)=s(t – t), where T = 1/2Wc, and
s(t) = sin(2pWc t)/ 2pWct
1.2
1
0.8
0.6
0.4
0.2
0 t
-7T -6T -5T -4T -3T -2T T
-1-0.2 0 1T 2T 3T 4T 5T 6T 7T

-0.4

l s(t) has zero crossings at t = kT, k = +1, +2, …

l Pulses can be packed every T seconds with zero interference
 Example of composite waveform
Three Nyquist pulses +s(t) +s(t-T)
1
shown separately
l + s(t)

l + s(t-T) 0 t
-2 T -1T 0 1T 2T 3T 4T
l - s(t-2T)

Composite waveform
-1
r(t) = s(t)+s(t-T)-s(t-2T) -s(t-2T)
r(t)
Samples at kT 2

r(0)=s(0)+s(-T)-s(-2T)=+1 1

r(T)=s(T)+s(0)-s(-T)=+1
0 t
r(2T)=s(2T)+s(T)-s(0)=-1 -2T -1T 0 1T 2T 3T 4T

Zero ISI at sampling -1

times kT -2
Digital Transmission
Fundamentals
Fundamental Limits in Digital
Transmission
Digital Binary Signal
1 0 1 1 0 1
+A

0 T 2T 3T 4T 5T 6T
-A

Bit rate = 1 bit / T seconds

For a given communications medium:
l How do we increase transmission speed?

l How do we achieve reliable communications?

l Are there limits to speed and reliability?

 Signaling with Nyquist Pulses
l p(t) pulse at receiver in response to a single input pulse (takes
into account pulse shape at input, transmitter & receiver filters,
and communications medium)
l r(t) waveform that appears in response to sequence of pulses
l If p(t) is a Nyquist pulse, then r(t) has zero intersymbol
interference (ISI) when sampled at multiples of T

1 0 1 1 0 1
+A
0 T 2T 3T 4T 5T t
-A

Transmitter Communication Receiver r(t)

Filter Medium Filter Receiver

Received signal
Pulse Transmission Rate
l Objective: Maximize pulse rate through a channel,
that is, make T as small as possible

Channel
T t t

l If input is a narrow pulse, then typical output is a

spread-out pulse with ringing
l Question: How frequently can these pulses be
transmitted without interfering with each other?
l Answer: 2 x Wc pulses/second

where Wc is the bandwidth of the channel

Multilevel Signaling
l Nyquist pulses achieve the maximum signalling rate with zero
ISI,
2Wc pulses per second or
2Wc pulses / Wc Hz = 2 pulses / Hz

l With two signal levels, each pulse carries one bit of

information
Bit rate = 2Wc bits/second

l With M = 2m signal levels, each pulse carries m bits

Bit rate = 2Wc pulses/sec. * m bits/pulse = 2Wc m bps

l Bit rate can be increased by increasing number of levels

l r(t) includes additive noise, that limits number of levels that
can be used reliably.
Example of Multilevel Signaling
l Four levels {-1, -1/3, 1/3, +1} for {00,01,10,11}
l Waveform for 11,10,01 sends +1, +1/3, -1/3
l Zero ISI at sampling instants

1.2

0.8
Composite waveform
0.6

0.4

0.2

0
-1 0 1 2 3
-0.2

-0.4

-0.6
 Noise Limits Accuracy
l Receiver makes decision based on transmitted pulse level + noise
l Error rate depends on relative value of noise amplitude and spacing
between signal levels
l Large (positive or negative) noise values can cause wrong decision
l Noise level below impacts 8-level signaling more than 4-level signaling

+A +A
+5A/7
+A/3 +3A/7
+A/7
-A/7
-A/3 -3A/7
Typical noise
-5A/7
-A -A

Four signal levels Eight signal levels

 Noise distribution
l Noise is characterized by probability density of amplitude samples
l Likelihood that certain amplitude occurs
l Thermal electronic noise is inevitable (due to vibrations of electrons)
l Noise distribution is Gaussian (bell-shaped) as below

s2 = Avg Noise Power

x
x0
Pr[X(t)>x0 ] = ? t

Pr[X(t)>x0 ] =
1 x2 2 2
e Area under
2 graph

0 x0 x
 Probability of Error
l Error occurs if noise value exceeds certain magnitude
l Prob. of large values drops quickly with Gaussian noise
l Target probability of error achieved by designing system so
separation between signal levels is appropriate relative to
average noise power

0 2 4 6 8
1.00E+00 d/2s
1.00E-01
1.00E-02
1.00E-03
1.00E-04
Pr[X(t)>d ] 1.00E-05
1.00E-06
1.00E-07
1.00E-08
1.00E-09
1.00E-10
1.00E-11
1.00E-12
Channel Noise affects Reliability

signal noise signal + noise

High
SNR

virtually error-free
signal noise signal + noise
Low
SNR
error-prone
Average Signal Power
SNR =
Average Noise Power

SNR (dB) = 10 log10 SNR

 Shannon Channel Capacity
l If transmitted power is limited, then as M increases spacing
between levels decreases
l Presence of noise at receiver causes more frequent errors
to occur as M is increased

Shannon Channel Capacity:

The maximum reliable transmission rate over an ideal channel
with bandwidth W Hz, with Gaussian distributed noise, and
with SNR S/N is
C = W log2 ( 1 + S/N ) bits per second

l Reliable means error rate can be made arbitrarily small by

proper coding
Example
l Consider a 3 kHz channel with 8-level signaling.
Compare bit rate to channel capacity at 20 dB SNR

l 3KHz telephone channel with 8 level signaling

Bit rate = 2*3000 pulses/sec * 3 bits/pulse = 18 kbps

l 20 dB SNR means 10 log10 S/N = 20

Implies S/N = 100
l Shannon Channel Capacity is then
C = 3000 log ( 1 + 100) = 19, 963 bits/second