Introduction to Digital Communicationsagullive/digitalintro.pdf · Introduction to Digital...

Introduction to Digital Communications

Aaron Gulliver

Dept. of Electrical and Computer Engineering

University of Victoria

Analog vs. Digital

• Analog signals– Value varies continuously

• Digital signals– Values limited to a finite set

• Binary signals– Two valued– Time T needed to send 1 bit– Data rate R=1/T bits per second

t

x(t)

t

x(t)

t

x(t) 1

0 0 0

1 1

0T

2Introduction to Digital Communications2008

Information Representation

• Communication systems must convert information into a form suitable for transmission

• Analog systemsAnalog signals are directly modulated

– AM, FM radio

• Digital systems Generate bits and transmit digital signals

– Computer communications, Cellular telephones

• Analog signals can be converted into digital signals


Digital Communication System

(a) Transmitter. (b) Receiver.


Digital Transmitter• Matches the message to the channel• If the message is analog, it must be sampled in time and

quantized in amplitude. – discrete signal in time and amplitude

• Encoder: – adds redundancy for error correction.

• Modulation encodes the message into the amplitude, phase or frequency of the carrier signal (PSK, FSK, QAM, OFDM, PAM, PPM)

• Advantages: – Reduces noise and interference– Multiplexing– Channel assignment

• Examples: television, radio, 802.11, cellphones, bluetooth, GPS, …

52008 Introduction to Digital Communications

Modulation

• Convert the digital information into waveforms suitable for the channel

• OOK

• PAM

0 1 1 0 1 0 0 1


Receiver

7

• Extracts the message from the received signal

• Operations: Filtering, Amplification, Demodulation

• The ideal receiver output is a scaled, delayed version of the message signal

• Decoder:

• estimates the original message from the received signal.


Channel

8

• Physical medium that that the signal is transmitted through

• Examples: Air, wires, coaxial cables, fiber optic cables

• Every channel introduces some amount of distortion, noise and interference

• The channel properties determine

• Data throughput of the system

• Quality of service (QoS) offered by the system


Noise and Interference

9

• Internal Noise

– Generated by components within a communication system (thermal noise)

• External Noise and Interference

– Atmospheric noise (electrical discharges)

– Man-made noise (ignition noise)

– Multipath interference (multiple transmission paths)

– Multiple access interference (signals from other users)


Advantages

• Many sources are digital in nature– Data, images, text, video, music

• Different sources can be treated the same• Flexibility

– Encryption– Compression (source coding)– Error correction/detection

• Reliable reproduction of signals - regeneration– Two states vs. infinite variety of shapes

• Greater immunity to noise and interference• Power efficient and spectral efficient


Introduction to Digital Communications 11

Digital versus Analog• Advantages of digital communications

– Regenerator receiver

– Different kinds of digital signals can be treated identically.

Data

Voice

Media

Propagation distance

Originalpulse

Regeneratedpulse

bits are bits!

2008

Digital Advantages

• Source coding compression algorithms can dramatically reduce the bit rate required to represent signals without significant distortion.

• Signal processing and channel coding techniques have significantly increased the bit rate that can be supported by a physical channel.

• Integrated circuits make complex signal processing and coding functions cost effective.


Disadvantages

• Complex signal processing• Synchronization problems• Non-graceful degradation in performance as

the SNR decreases

2008 Introduction to Digital Communications 13

Performance Metrics• In analog communications we want

• In digital communications– Data rate (R bps) (limited by the Channel Capacity)

– Resources consumed: bandwidth, power

– Quality of the communications link : typically measured in terms of the Bit Error Rate (BER) or probability of error, PE

• Number of bit errors that occur for a given number of bits transmitted.

• Optical channels: Pe = 10-9

• Wireless channels: voice Pe = 10-3 data Pe = 10-6

– Propagation and processing delay

– Timing jitter in the bitstream at the receiver

)()(ˆ tmtm


Applications

• Internet (last mile, VOIP)

• Local and long distance telephone channels

• Fibre optics (backbone and fibre to the home)

• Satellite communications (HDTV)

• CDs and DVDs

• Digital audio (mp3)

• Wireless Communications

• Cellular Communications

– GSM, TDMA, FDMA, CDMA, 3G

• Wireless LANs (802.11)

• WiMAX

• Bluetooth (headsets)

• Cordless telephones


Goals of Digital Communications Design

• Maximize bit rate R

• Minimize probability of error PE

• Minimize required signal-to-noise ratio (SNR)

• Minimize required bandwidth W

• Maximize system utilization Capacity

• Minimize system complexity

• Minimize cost $


Span of Wireless Networks


18

GSM Mobile Phone

Transmitter

Receiver

19

CDMA Cell Phone

Mobile phone transmitter

Baseband Data Transmission - PAM

System model and waveforms

for synchronous baseband

digital data transmission.

(a) Baseband digital data

communication system.

(b) Typical transmitted

sequence. (c) Received

sequence plus noise.


• Each T second pulse represents a bit of data

• Receiver has to decide whether a 1 or 0 was received (A or –A)

• Integrate-and-dump detector


Receiver Structure

Receiver structure and integrator output. (a) Integrate-and-dump receiver. (b) Output from the integrator.


Receiver Preformance• The output of the integrator is

V is a random variable• N is Gaussian noise

0

0

[ ( ) ( )]

is sent

is sent

t T

t

V s t n t dt

AT N A

AT N A

0

0

( )

t T

t

N n t dt


2005-01-21 Lecture 1 24

Noise in communication systems

Thermal noise is described by a zero-mean Gaussian random process, n(t).

Its PSD is flat, hence, it is called white noise

N0/2 W/Hz

Probability density function

Power spectral density

2 2/(2 )

2( )

2

n

N

ef n

Analysis

since AWGN is uncorrelated

2)(

2

)]()([

)(][

][][][

0)]([])([][

00

2

2

22

0

0

0

0

0

0

0

0

0

0

0

0

0

0

TNdtdsst

N

dtdssntnE

dttnENE

NENENVar

dttnEdttnENE

Tt

t

Tt

t

Tt

t

Tt

t

Tt

t

Tt

t

Tt

t


Error Analysis

• The pdf of N is

• In how many different ways can an error occur?

TN

enf

TNn

N

0

)/( 02

)(


Error Analysis• Two ways in which errors occur

– A is transmitted, AT+N<0 (0 received,1 sent)

– -A is transmitted, -AT+N>0 (1 received,0 sent)

Error probabilities for binary signaling.


• Similarly

• The average probability of error is

20/ 2

00

2( | )

AT n N Te A T

P Error A dn QNN T

20/ 2

00

2( | )

n N T

AT

e A TP Error A dn Q

NN T

0

22

)()|()()|(

N

TAQ

APAEPAPAEPPE


• Energy per bit

• Therefore, PE can be written in terms of the energy.

• Define

TAdtAE

Tt

t

b

220

0

00

2

N

E

N

TAz b


• Recall: Rectangular pulse of duration Tseconds has magnitude spectrum

• Effective Bandwidth

• Therefore

TBp /1

pBN

Az

0

2

sinc( )AT Tf


Probability of Error vs. SNR

Pe for antipodal

baseband digital

signaling.


Probability of Error Approximation

• Use the approximation

1,2

2

1,2

)(

0

2

2/2

zz

e

N

TAQP

uu

euQ

z

E

u


Example

• Digital data is transmitted through a baseband system with , the received pulse amplitude is A = 20mV.

a) If the transmission rate is 1kbps, what is the probability of error?

3

3

2 62

7 3

0

3

1 110

10

400 10400 10 4V

10 10

2.58 102

p

p

z

E

BT

ASNR z

N B

eP

z

7

0 10 W/HzN


b) If 10 kbps are transmitted, what must the value of A be to attain the same probability of error?

• Conclusion: tradeoff is

Transmission power vs. Bit rate

2 22 3

7 4

0

4 4 10 63.2mV10 10p

A Az A A

N B


Bandpass Modulation

• There are 3 parameters

– Amplitude A(t) ― Amplitude Modulation

– Frequency f(t) ― Frequency Modulation

– Phase φ(t) ― Phase Modulation

V(t) = A cos(2πfc t + Φ)

Binary Signaling Techniques

Waveforms forASK, PSK, and FSK modulation.


ASK, PSK, and FSK

• Amplitude Shift Keying (ASK)

• Phase Shift Keying (PSK)

• Frequency Shift Keying

0)(0

1)()2cos()2cos()()(

b

bcc

ccnTm

nTmtfAtfAtmts

1)()2cos(

1)()2cos()2cos()()(

bcc

bcc

ccnTmtfA

nTmtfAtftmAts

1)()2cos(

1)()2cos()(

2

1

bc

bc

nTmtfA

nTmtfAts

1 0 1 1

1 0 1 1

1 0 1 1

AM Modulation

PM Modulation

FM Modulation

m(t)

m(t)


Amplitude Shift Keying (ASK)

What is the structure of the optimum receiver?

1 cos(2 )cA f t

0 0


Receiver for Binary Signals in Noise

Receiver structure for detecting binary signals in additive white Gaussian noise (AWGN)


Error Analysis

• 0s1(t), 1s2(t)

• Received signal:

• Noise is white and Gaussian

• Find PE

– In how many different ways can an error occur?

1 0 0

2 0 0

( ) ( ) ( ),

or

( ) ( ) ( ),

y t s t n t t t t T

y t s t n t t t t T


Error Analysis (General Case)• Two types of errors:

• Receive 1 Send 0

• Receive 0 Send 1

• Decision process: • The received signal is filtered

• Filter output is sampled every T seconds

• Threshold k

• An error occurs when:

kTnTsTv

kTnTsTv

)()()(

or

)()()(

002

001


• are filtered signal and noise terms.

• Noise term: is filtered white Gaussian noise.– therefore it is Gaussian

• The PSD is

– mean zero

– variance is equal to the average power of the noise process

00201 ,, nss

20 )(2

)(0

fHN

fSn

dffHN 202 )(2

( )on t


• The pdf of the noise term is

• Note that we still don’t know what the filter is.

• Will any filter work? Or is there an optimal one?

• Recall that in the baseband case (no modulation), we used an integrator

– equivalent to filtering with

2

2/

2)(

022

n

N

enf

fjfH

2

1)(


• The input to the threshold device is

• These are also Gaussian random variables

– mean:

– variance: same as the variance of N

01

02

( ) ( )

or

( ) ( )

V v T s T N

V v T s T N

01 02( ) or ( )s T s T


Distribution of V

• The distribution of V, the input to the threshold device is

Conditional probability density functions of the filter output at time t = T


Probability of Error

• Two types of errors

• The average probability of error

)(1

2))(|(

)(

2))(|(

02

2

2/)]([

2

01

2

2/)]([

1

2202

2201

TskQdv

etsEP

TskQdv

etsEP

k Tsv

k

Tsv

)](|[2

1)](|[

2

121 tsEPtsEPPE


• Goal: Minimize the average probability of error

– choose the optimal threshold

• What should the optimal threshold, kopt be?

– kopt=0.5[s01(T)+s02(T)]

2

)()( 0102 TsTsQPE


Observations

• PE is a function of the difference between the two signals.

• Recall: Q-function decreases with increasing argument.

• Therefore, PE will decrease with increasing distance between the two output signals

• Choose the filter h(t) such that PE is a minimum– maximize the difference between the two signals at the

output of the filter


Matched Filter

• Goal: Given , choose H(f) such that

is maximized.

• The solution to this problem is known as the matched filter and is given by

• Therefore, the optimum filter depends on the input signals.

)(),( 21 tsts

)()( 0102 TsTsd

)()()( 120 tTstTsth


Matched Filter Receiver

Matched filter receiver for binary signaling in additive white Gaussian noise (AWGN).


Error Probability for Matched Filter Receiver

• Recall

• The maximum value of the distance is

• E1 is the energy of the first signal

• E2 is the energy of the second signal

2

dQPE

)2(2

122121

0

2

max EEEEN

d

dttstsEE

)()(1

21

21

12

51Introduction to Digital Communications

0 0

0 0

2 2

1 1 2 2( ) ( )

t T t T

t t

E s t dt E s t dt

2008

• Therefore

• Probability of error depends on the signal energies (just as in the baseband case), noise power, and the similarity between the signals.

• If we make the transmitted signals as dissimilar as possible, then the probability of error will decrease.

• This is achieved with

2/1

0

122121

2

2

N

EEEEQPE

112


ASK

• The matched filter:

• Optimum Threshold:

• Similarity between signals?

• Therefore

• 3dB worse than baseband.

)2cos()(,0)( 21 tfAtsts c

)2cos( tfA c

TA2

4

1

zQN

TAQPE

0

2

4


PSK

• Modulation index: m (determines the phase shift)

• Matched Filter with threshold 0

• For m = 0, 3dB better than ASK

)cos2sin()(

)cos2sin()(

1

2

1

1

mtfAts

mtfAts

c

c

)2cos(12 2 tfmA c

))1(2( 2 zmQPE


Matched Filter for PSK

Optimum correlation receiver for PSK.


FSK

•

• Probability of error:

• Same as ASK

))(2cos()(

)2cos()(

2

1

tffAts

tfAts

c

c

T

mf

)( zQ


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