Digital Communication System

Source coding

• Represent signals into digital data (e.g. bits)– E.g. video, audio

• How to use as little amount of digital data as possible without losing any information

• How to use as little amount of digital data as possible with an acceptable loss of information (fidelity)

Channel codingAdding redundancy to correct error; e.g.,

Modulation

modulator channel demodulatorSourceSignal

Bit stream

Dest.

bit

+

n(t)

Converting bits to waveforms

e.g., Digital modulation

Intuitive picture

Naive

Digital Modulation examples

1 1 1 10 0 0

t

ON-OFF keyinge.g. optical fiber

01 01 11 00 1100 m001 m110 m211 m3

Bits Symbol

Amplitude Shift Keying (ASK)

Ts

m0

m1

m2

m3

Bit duration vs Symbol duration

Frequency Shift Keying (FSK)

Ts

m0

m1

( )[ ]0cos 2 cA f t f tπ + ∆

( )[ ]1cos 2 cA f t f tπ + ∆

mi( )[ ]cos 2 c iA f t f tπ + ∆

Phase Shift Keying (PSK)

{ }

( )

0 1 1, , ,

cos 2

i M

i c i

m m m m

m A f tπ θ

−∈

= +

tTs

different Phase

Quadrature PSK

( )[ ] ( ) ( )[ ] ( )

( )( )( )( )

0 0

1 1

2 2

3 3

: cos(2 ) cos cos 2 sin sin 2

1 1(0, 0) : , ,

4 2 23 1 1

(0,1) : , ,4 2 25 1 1

(1,1) : , ,4 2 27 1 1

(1, 0) : , ,4 2 2

i c i i c i cm f t f t f t

m

m

m

m

π θ θ π θ π

πθ

πθ

πθ

πθ

+ = −

=

−=

− −=

−=

QPSK modulator

X

X

+ channel

I1

S2

=

Q1

S2

−=

( )cos 2 cf tπ

( )sin 2 cf tπ−

m3 :

( )

( ) ( )( )

1cos 2

21

sin 22

c

c

f t

f t

π

π−

+ −

r(t)

QPSK Demodulator

( )

0

sT

dt∫ i

( )

0

sT

dt∫ i

X

X

( )cos 2 cf tπ

( )sin 2 cf tπ−

( ) ( )

( )

( )

cos 2

sin 2

I c

Q c

r t S f t

S f t

n t

π

π

=

−

+

IS2s

IT

n+

QS2s

QT

n+

( ) ( )

( ) ( )( ) ( ) ( ) ( )2I Q

cos 2

S cos 2 +S sin 2 cos 2 cos 2

c

c c c c

r t f t

f t f t f t n t f t

π

π π π π= − +

1integer multiple of sc

T f=

note

( ) ( ) ( )

( )( )

( )( )

( )

2

2

2

0

1sin 2 cos 2 sin 2 2

2

1 cos 2 2cos 2

2

1 cos 2 2sin 2

2

cos 22

s

c c c

cc

cc

Ts

c

f t f t f t

f tf t

f tf t

Tf t dt

π π π

ππ

ππ

π

⋅ = ⋅

+ ⋅=

− ⋅=

=∫

8PSK

001 110 101 111

000 m0001 m1

111 m7

I-Q representation of QPSK

( ) ( ) ( ) ( ) ( )( ): cos cos 2 sin sin 2i X i c i cm S t f t f tθ π θ π= + −

( )( )( )( )

0

1

2

3

1 1: ,

2 21 1

: ,2 21 1

: ,2 21 1

: ,2 2

m

m

m

m

−

− −

−

m0m1

m3m2

12

12

( )cos 2 cf tπ

( )sin 2 cf tπ−

In-phasecomponent

Quadrature-phasecomponent

I-Q representation of 8PSK

m0

m1

m4

m3

12

( )cos 2 cf tπ

( )sin 2 cf tπ−

In-phasecomponent

Quadrature-phasecomponent

m2

m5

m6

m7

“Symbol constellation”“Signal constellation”

Amplitude-Phase Shift Keying (M-QAM)

Carrier acquisition and tracking

• Carrier frequency can drift.• Moreover, there is a Doppler effect in

mobile stations.

Symbol time synchronization and phase coherency

• It is done on the basis of the discontinuity of the waveforms between adjacent symbols.

• If there is no channel distortion, the delineated boundary between symbols and the shape of the waveform would give the phase of the waveforms for each symbol

• In real-life there is distortion, so the phase information is not given even if the symbol interval is delineated. For coherent detection, the phase information must be provided through additional circuits.

Imperfect Channel and Noise

• E.g. Pair of copper wire, coaxial cable, optical fiber, wireless channel

Band-limited channel

• Example of Bertsekas and Gallager, “Data Networks” 1992. Fig. 2.3

– Low pass filter

channel

Distortion of band-limited channel.Introduces inter-symbol interference (ISI).

Note

A rectangular wave’s bandwidth is large.The band-limited channel clips the high-frequency component of the signal.

Channel model

• Channel take input signals and gives rise to output signals; can be viewed as a “system” or “filter”.

Inter Symbol Interference (ISI)

• Perfect Channel

( ) ( )( ) 1

h t t

H f

δ=

=( )x t ( )x t

channel

• In real world,

|H(f)|Signal distortion can occur!

Inter Symbol Interference (ISI)

• Example: NRZ (Non-Return to Zero) code

t

channel

tT T

Single Pulse:

Smear to next symbol duration.

Result in distortion (ISI).

channel

NRZ Pulse Train:

Same story (distortion) in the pass-band signal.

1 1 1 10 0 0

t

ASK

f

Distortion in QAM

X

X

+ channel

IS ( )n

QS ( )n

( )cos 2 cf tπ

( )sin 2 cf tπ−

p(t-nT)

p(t-nT)

In the previous example, p(t) was a rectangular wave.

f

Theoretical guidance: Nyquistpulse

• In fact, by designing p(t) to be other than the rectangular pulse and designing a receiver filter, one can avoid ISI.

• Solution to ISI: Equalizer

H(f)

Channel Receiver

Equalizer

S(t) S(t)

In real world, delay is considered: S(t-d)

H-1(f) as an equalizer?Noise is greatly amplified at frequencies with low |H(f)|.