Fundamentals of Digital Communication

2

Digital communication system

Low Pass Filter

Sampler Quantizer Channel Encoder

Line Encoder

Pulse Shaping

Filters

SourceEncoder

Modulator

MultiplexerInputSignalAnalog/Digital

To Channel

DetectorReceiverFilter

De-ModulatorFrom Channel

Channel Decoder

Digital-to-AnalogConverter

De-Multiplexer

Signalat the user end

Carrier

Carrier Ref.

3

Noiseless Channels and Nyquist Theorem

For a noiseless channel, Nyquist theorem gives the relationship between the channel bandwidth and maximum data rate that can be transmitted over this channel.

mBC 2log2

Nyquist Theorem

C: channel capacity (bps)B: RF bandwidthm: number of finite states in a symbol of transmitted signal

Example: A noiseless channel with 3kHz bandwidth can only transmit a maximum of 6Kbps if the symbols are binary symbols.

4

Nyquist minimum bandwidth requirement

The theoretical minimum bandwidth needed for baseband transmission of Rs symbols per second is Rs/2 hertz ?

t

)/sinc()( Ttth 1

0 T T2TT2

T21

T21

T

)( fH

f0

5

Shannon’s Bound for noisy channels

There is a fundamental upper bound on achievable bandwidth efficiency.

Shannon’s theorem gives the relationship between the channel bandwidth and the maximum data rate that can be transmitted over a noisy channel .

)1(log2max NS

BC

B

Shannon’s Theorem

C: channel capacity (maximum data-rate) (bps)B or W : RF bandwidthS/N: signal-to-noise ratio (no unit)

6

Shannon limit … Shannon theorem puts a limit on

transmission data rate, not on error probability:

Theoretically possible to transmit information at any rate Rb , where Rb C with an arbitrary small error probability by using a sufficiently complicated coding scheme.

For an information rate Rb > C , it is not possible to find a code that can achieve an arbitrary small error probability.

7

Shannon limit …C/W

[bits

/s/H

z]

SNR [dB]

Practical region

Unattainableregion

8

Shannon limit …

There exists a limiting value of below which there can be no error-free communication at any information rate.

By increasing the bandwidth alone, the capacity cannot be increased to any desired value.

WC

NE

WC b

02 1log

WNNCES

NSWC

b

0

2 1log

[dB] 6.1693.0log

1

:get we,0or As

20

eNE

WCW

b

0/ NEb

Shannon limit

9

Shannon limit …

[dB] / 0NEb

W/C [Hz/bits/s] Practical region

Unattainableregion

-1.6 [dB]

10

Bandwidth efficiency plane

R<CPractical region

R>CUnattainable region

R/W

[bits

/s/H

z]

Bandwidth limited

Power limited

R=C

Shannon limit510BP

MPSKMQAMMFSK

M=2

M=4

M=8

M=16

M=64

M=256

M=2M=4

M=8

M=16

[dB] / 0NEb

11

Error probability plane(example for coherent MPSK and MFSK)

[dB] / 0NEb [dB] / 0NEb

Bit

erro

r pro

babi

lity

M-PSK M-FSK

k=1,2

k=3

k=4

k=5

k=5

k=4

k=2

k=1

bandwidth-efficient power-efficient

12

M-ary signaling Bandwidth efficiency:

Assuming Nyquist (ideal rectangular) filtering at baseband, the required passband bandwidth is:

M-PSK and M-QAM (bandwidth-limited systems) Bandwidth efficiency increases as M increases.

MFSK (power-limited systems) Bandwidth efficiency decreases as M increases.

][bits/s/Hz 1log2

bs

b

WTWTM

WR

[Hz] /1 ss RTW

][bits/s/Hz log/ 2 MWRb

][bits/s/Hz /log/ 2 MMWRb

13

Power and bandwidth limited systems

Two major communication resources: Transmit power and channel bandwidth

In many communication systems, one of these resources is more precious than the other. Hence, systems can be classified as:

Power-limited systems: save power at the expense of bandwidth (for example by using coding schemes)

Bandwidth-limited systems: save bandwidth at the expense of power (for example by using spectrally efficient modulation

schemes)

14

Goals in designing a DCS Goals:

Maximizing the transmission bit rate Minimizing probability of bit error Minimizing the required power Minimizing required system

bandwidth Maximizing system utilization Minimize system complexity

15

Limitations in designing a DCS The Nyquist theoretical minimum bandwidth

requirement The Shannon-Hartley capacity theorem (and

the Shannon limit) Government regulations Technological limitations Other system requirements (e.g satellite

orbits)