DC Digital Communication PART7

8/7/2019 DC Digital Communication PART7

1/200

Modulation, Demodulation and

Coding Course

Period 3 - 2005

Sorour Falahati

Lecture 3


2/200

2005-01-26 Lecture 3 2

Last time we talked about:

Transforming the information source to aform compatible with a digital system

Sampling

Aliasing

Quantization

Uniform and non-uniform

Baseband modulation

Binary pulse modulation M-ary pulse modulation

M-PAM (M-ay Pulse amplitude modulation)


3/200

2005-01-26 Lecture 3 3

Formatting and transmission of basebandsignal

Information (data) rate:

Symbol rate : For real time transmission:

Sampling at rate

(sampling time=Ts)

Quantizing each sampled

value to one of theL levels in quantizer.

Encoding each q. value tobits

(Data bit duration Tb=Ts/l)

Encode

PulsemodulateSample Quantize

Pulse waveforms(baseband signals)

Bit stream(Data bits)

Format

Digital info.

Textualinfo.

Analoginfo.

source

Mapping every data bits to a

symbol out of M symbols and transmittinga baseband waveform with duration T

ss Tf /1! Ll 2log!

Mm 2log!

[bits/sec]/1 bbR !ec][symbols/s/1 TR !

mRRb

!


4/200

2005-01-26 Lecture 3 4

Qunatization example

t

Ts: samplingtime

x(

nTs): sampledvaluesxq(nTs): quantizedvalues

boundaries

Quant. levels

111 3.1867

110 2.2762

101 1.3657

100 0.4552

011 -0.4552

010 -1.3657

001 -2.2762

000 -3.1867

PCM

codeword 110 110 111 110 100 010 011 100 100 011 PCM sequence

amplitudex(t)


5/200

2005-01-26 Lecture 3 5

Example of M-ary PAM

0 Tb 2Tb 3Tb 4Tb 5Tb 6Tb

0 Ts 2Ts

0T 2T 3T

2.2762 V 1.3657 V

1 1 0 1 0 1-B

B

T

01

3B

T

T

-3B

T

0010

1

A.

T

0

T

-A.

Assuming real time tr. and equal energy per tr. data bit forbinary-PAM and 4-ary PAM:

4-ary: T=2Tb and Binay: T=Tb

4-ary PAM(rectangular pulse)

Binary PAM(rectangular pulse)

11

0 T 2T 3T 4T 5T 6T

22 10BA !


6/200

2005-01-26 Lecture 3 6

Today we are going to talk about:

Receiver structure Demodulation (and sampling)

Detection

First step for designing the receiver Matched filter receiver

Correlator receiver

Vector representation of signals (signal

space), an important tool to facilitate Signals presentations, receiver structures

Detection operations


7/200

2005-01-26 Lecture 3 7

Demodulation and detection

Major sources of errors: Thermal noise (AWGN)

disturbs the signal in an additive fashion (Additive)

has flat spectral density for all frequencies of interest (White)

is modeled by Gaussian random process (Gaussian Noise)

Inter-Symbol Interference (ISI) Due to the filtering effect of transmitter, channel and receiver,

symbols are smeared.

Format Pulsemodulate

Bandpassmodulate

Format DetectDemod.

& sample

)(tsi)(tgiim

im )(tr)(Tz

channel)(th

c

)(tn

transmitted symbol

estimated symbol

Mi ,,1 -!M-ary modulation


8/200

2005-01-26 Lecture 3 8

Example: Impact of the channel


9/200

2005-01-26 Lecture 3 9

Example: Channel impact

)75.0(5.0)()( Tttthc

! HH


10/200

2005-01-26 Lecture 3 10

Receiver job

Demodulation and sampling: Waveform recovery and preparing the

received signal for detection:

Improving the signal power to the noise power

(SNR) using matched filter

Reducing ISI using equalizer

Sampling the recovered waveform

Detection: Estimate the transmitted symbol based on

the received sample


11/200

2005-01-26 Lecture 3 11

Receiver structure

Frequencydown-conversion

Receivingfilter

Equalizingfilter

Thresholdcomparison

For bandpass signals Compensation forchannel induced ISI

Baseband pulse(possibly distored)

Sample(test statistic)

Baseband pulseReceived waveform

Step 1 waveform to sample transformation Step 2 decision making

)(tr )(Tz im

Demodulate & Sample Detect


12/200

2005-01-26 Lecture 3 12

Baseband and bandpass

Bandpass model of detection process isequivalent to baseband model because:

The received bandpass waveform is firsttransformed to a baseband waveform.

Equivalence theorem:

Performing bandpass linear signal processingfollowed by heterodying the signal to thebaseband, yields the same results as

heterodying the bandpass signal to thebaseband , followed by a baseband linear signalprocessing.


13/200

2005-01-26 Lecture 3 13

Steps in designing the receiver

Find optimum solution for receiver designwith the following goals:1. Maximize SNR

2. Minimize ISI

Steps in design: Model the received signal

Find separate solutions for each of the goals.

First, we focus on designing a receiver

which maximizes the SNR.


14/200

2005-01-26 Lecture 3 14

Design the receiver filter to maximizethe SNR

Model the received signal

Simplify the model:

Received signal in AWGN

)(thc

)(tsi

)(tn

)(tr

)(tn

)(tr)(tsiIdeal channels)()( tthc

H!

AWGN

AWGN

)()()()( tnthtstrci !

)()()( tntstr i !


15/200

2005-01-26 Lecture 3 15

Matched filter receiver

Problem: Design the receiver filter such that the SNR is

maximized at the sampling time when

is transmitted.

Solution: The optimum filter, is the Matched filter, given by

which is the time-reversed and delayed version of theconjugate of the transmitted signal

)(th

)()()(*

tTsthth iopt !!)2exp()()()(

*fTjfSfHfH iopt T!!

Mitsi ,...,1),( !

T0 t

)(tsi

T0 t

)()( thth opt!


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2005-01-26 Lecture 3 16

Example of matched filter

T t T t T t0 2T

)()()( thtsty opti !2A)(tsi )(thopt

T t T t T t0 2T

)()()( thtsty opti !2A)(tsi )(thopt

T/2 3T/2T/2 T T/2

2

2TA

T

A

T

A

T

A

T

A

T

A

T

A


17/200

2005-01-26 Lecture 3 17

Properties of the matched filter1. The Fourier transform of a matched filter output with the matched

signal as input is, except for a time delay factor, proportional to theESD of the input signal.

2. The output signal of a matched filter is proportional to a shiftedversion of the autocorrelation function of the input signal to whichthe filter is matched.

3. The output SNR of a matched filter depends only on the ratio of thesignal energy to the PSD of the white noise at the filter input.

4. Two matching conditions in the matched-filtering operation: spectral phase matching that gives the desired output peak at time T.

spectral amplitude matching that gives optimum SNR to the peak value.

)2exp(|)(|)( 2 fTjfSfZ T!

sss ERTzTtRtz !!! )0()()()(

2/max

0N

E

N

S s

T

!


18/200

2005-01-26 Lecture 3 18

Correlator receiver

The matched filter output at thesampling time, can be realized as thecorrelator output.

"!!

!

)(),()()(

)()()(

*

0

tstrdsr

TrThTz

i

T

opt

XXX


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2005-01-26 Lecture 3 19

Implementation of matched filterreceiver

Mz

z

/

1

z!

)(tr

)(1 Tz)(

*

1 tTs

)(*

tTsM

)(TzM

z

Bank of M matched filters

Matched filter output:Observation

vector

)()( tTstrz ii ! Mi ,...,1!

),...,,())(),...,(),(( 2121 MM zzzTzTzTz !!z


20/200

2005-01-26 Lecture 3 20

Implementation of correlator receiver

dttstrz i

T

i )()(0

!

T

0

)(1 ts

T

0

)(ts M

Mz

z

/

1

z!)(tr

)(1 Tz

)(TzM

z

Bank of M correlators

Correlators output:Observation

vector

),...,,())(),...,(),(( 2121 MM zzzTzTzTz !!z

Mi ,...,1!


21/200

2005-01-26 Lecture 3 21

Example of implementation ofmatched filter receivers

2

1

z

zz!

)(tr

)(1 Tz

)(2 Tz

z

Bank of 2 matched filters

T t

)(1 ts

T t

)(2 tsT

T0

0

T

A

T

AT

A

T

A

0

0


22/200

2005-01-26 Lecture 3 22

Signal space

What is a signal space? Vector representations of signals in an N-

dimensional orthogonal space

Why do we need a signal space? It is a means to convert signals to vectors and vice

versa.

It is a means to calculate signals energy andEuclidean distances between signals.

Why are we interested in Euclidean distances

between signals? For detection purposes: The received signal is

transformed to a received vectors. The signal whichhas the minimum distance to the received signal isestimated as the transmitted signal.


23/200

2005-01-26 Lecture 3 23

Schematic example of a signal space

),()()()(

),()()()(

),()()()(

),()()()(

212211

323132321313

222122221212

121112121111

zztztztz

aatatats

aatatats

aatatats

!!

!!

!!

!!

z

s

s

s

]]

]]

]]

]]

)(1 t]

)(2 t]

),( 12111 aa!s

),( 22212 aa!s

),( 32313 aa!s

),( 21 zz!z

Transmitted signalalternatives

Received signal atmatched filter output


24/200

2005-01-26 Lecture 3 24

Signal space

To form a signal space, first we need toknow the inner product between twosignals (functions):

Inner (scalar) product:

Properties of inner product:

g

g

"! dttytxtytx )()()(),( *

= cross-correlation between x(t) and y(t)

""! )(),()(),( tytxatytax

""! )(),()(),( * tytxataytx

"""! )(),()(),()(),()( tztytztxtztytx


25/200

2005-01-26 Lecture 3 25

Signal space contd

The distance in signal space is measure bycalculating the norm.

What is norm? Norm of a signal:

Norm between two signals:

We refer to the norm between two signals asthe Euclidean distance between two signals.

xEdttxtxtxtx !!"! g

g2)()(),()(

)()( txatax !

)()(, tytxd yx !

= length of x(t)


26/200

2005-01-26 Lecture 3 26

Example of distances in signal space

)(1

t]

)(2 t]),( 12111 aa!s

),( 22212 aa!s

),( 32313 aa!s

),( 21 zz!z

zsd ,1

zsd ,2zsd

,3

The Euclidean distance between signalsz(t) ands(t):

3,2,1

)()()()( 2222

11,

!

!!

i

zazatztsd iiizsi

1E

3E

2E


27/200

2005-01-26 Lecture 3 27

Signal space - contd

N-dimensional orthogonal signal space ischaracterized by N linearly independent functions

called basis functions. The basis functionsmust satisfy the orthogonality condition

where

If all , the signal space is orthonormal.

_ aNjj

t1

)(!

]

jiij

T

iji Kdttttt H]]]] !"! )()()(),( *0

Tt ee0

Nij ,...,1, !

{p

!p

! ji

ji

ij 0

1

H

1!iK


28/200

2005-01-26 Lecture 3 28

Example of an orthonormal basisfunctions

Example: 2-dimensional orthonormal signal space

Example: 1-dimensional orthonornal signal space

1)()(

0)()()(),(

0)/2sin(2

)(

0)/2cos(2

)(

21

2

0

121

2

1

!!

!"!

e!

e!

tt

dttttt

TtTtT

t

TtTtT

t

T

]]

]]]]

T]

T]

T t

)(1 t]

T

1

0

)(1 t]

)(2 t]

0

1)(1 !t] )(1 t]0


29/200

2005-01-26 Lecture 3 29

Signal space contd

Any arbitrary finite set of waveformswhere each member of the set is of duration T,can be expressed as a linear combination of N

orthonogal waveforms where .

where

_ aM

ii ts 1)( !

_ aNjj

t1

)(!

] MN e

!

!N

j

jiji tats1

)()( ] Mi ,...,1!MN e

dtttsttsa

T

ji

j

ji

j

ij )()(1)(),(10

*

"!! ]] Tt ee0Mi ,...,1! Nj ,...,1!

),...,,( 21 iNiii aaa!s2

1

ij

N

j

ji aE !

!Vector representation of waveform Waveform energy


30/200

2005-01-26 Lecture 3 30

Signal space - contd

!

!N

j

jiji tats1

)()( ] ),...,,( 21 iNiii aaa!s

iN

i

a

a

/

1

)(1 t]

)(tN

]

1ia

iNa

)(tsi

T

0

)(1 t]

T

0

)(tN

]

iN

i

a

a

/

1

ms!)(tsi

1ia

iNa

ms

Waveform to vector conversion Vector to waveform conversion


31/200

2005-01-26 Lecture 3 31

Example of projecting signals to anorthonormal signal space

),()()()(

),()()()(

),()()()(

323132321313

222122221212

121112121111

aatatats

aatatats

aatatats

!!

!!

!!

s

s

s

]]

]]

]]

)(1 t]

)(2 t]

),( 12111 aa!s

),( 22212 aa!s

),( 32313 aa!s

Transmitted signalalternatives

dtttsa

T

jiij

)()(0

! ]Ttee0

M

i,...,1!Nj ,...,1!


32/200

2005-01-26 Lecture 3 32

Signal space contd

To find an orthonormal basis functions for a givenset of signals, Gram-Schmidt procedure can beused.

Gram-Schmidt procedure: Given a signal set , compute an orthonormal

basis1. Define

2. For compute

If let

If , do not assign any basis function.

3. Renumber the basis functions such that basis is

This is only necessary if for any i in step 2.

Note that

_ aMii

ts1

)(!_ aN

jjt

1)(

!]

)(/)(/)()( 11111 tstsEtst !!]

Mi ,...,2!

!

"!1

1

)()(),()()(i

j

jjiii tttststd ]]

0)( {tdi )(/)()( tdtdt iii !]

0)( !tdi

_ a)(),...,(),( 21 ttt N]]]

0)( !tdiMN e


33/200

2005-01-26 Lecture 3 33

Example of Gram-Schmidt procedure

Find the basis functions and plot the signal space forthe following transmitted signals:

Using Gram-Schmidt procedure:

T t

)(1 ts

T t

)(2 ts

)()(

)()(

)()(

21

12

11

AA

tAts

tAts

!!

!

!

ss

]

]

)(1 t]-A A0

1s2s

T

A

T

A

0

0

T t

)(1 t]

T

1

0

0)()()()(

)()()(),(

/)(/)()(

)(

122

01212

1111

0

22

11

!!

!"!

!!

!!

tAtstd

Adtttstts

AtsEtst

AdttsE

T

T

]

]]

]

1

2


34/200

2005-01-26 Lecture 3 34

Implementation of matched filter receiver

)(tr

1z)(1 tT ]

)( tTN ]N

z

Bank of N matched filters

Observationvector

)()( tTtrz jj ! ] Nj ,...,1!

),...,,( 21 Nzzz!z

!!

N

j

jiji tats1

)()( ]

MN e

Mi ,...,1!

Nz

z1

z!

z


35/200

2005-01-26 Lecture 3 35


),...,,( 21 Nzzz!z

Nj ,...,1!dtttrz jT

j )()(0

]!

T

0

)(1 t]

T

0

)(tN]

Nr

r

/

1

z!

)(tr

1z

Nz

z

Bank of N correlators

Observation

vector

!!

N

jjiji tats

1)()( ] Mi ,...,1!

MN e


36/200

2005-01-26 Lecture 3 36

Example of matched filter receivers usingbasic functions

Number of matched filters (or correlators) is reduced by 1 compared tousing matched filters (correlators) to the transmitted signal.

T t

)(1 ts

T t

)(2 ts

T t

)(1 t]

T

1

0

? A1z z!)(tr z

1 matched filter

T t

)(1 t]

T

1

0

1z

T

A

T

A

0

0


37/200

2005-01-26 Lecture 3 37

White noise in orthonormal signal space

AWGN n(t) can be expressed as)(~)()( tntntn !

Noise projected on the signal space

which impacts the detection process.

Noise outside on the signal space

"! )(),( ttnn jj ]

0)(),(~ "! ttn j]

)()(1

tntnN

j

jj!

! ]

Nj ,...,1!

Nj ,...,1!

Vector representation of

),...,,( 21 Nnnn!n

)( tn

independent zero-meanGaussain random variables withvariance

_ aN

jjn 1!

2/)var( 0Nnj !


38/200

Digital Communications I: Modulation

and Coding Course

Period 3 - 2007

Catharina LogothetisLecture 4


39/200

Lecture 4 39

Last time we talked about:

Receiver structure Impact of AWGN and ISI on the transmitted

signal

Optimum filter to maximize SNR

Matched filter receiver and Correlator receiver


40/200

Lecture 4 40

Receiver job

Demodulation and sampling: Waveform recovery and preparing the received

signal for detection:

Improving the signal power to the noise power (SNR)

using matched filter

Reducing ISI using equalizer

Sampling the recovered waveform

Detection:

Estimate the transmitted symbol based on the

received sample


41/200

Lecture 4 41

Receiver structure

Frequency

down-conversion

Receiving

filter

Equalizing

filter

Thresholdcomparison

For bandpass signals Compensation for

channel induced ISI

Baseband pulse

(possibly distored)Sample

(test statistic)Baseband pulse

Received waveform

Step 1 waveform to sample transformation Step 2 decision making

)(tr)(Tz

im

Demodulate & Sample Detect


42/200

Lecture 4 42


Mz

z

/

1

z!)(tr

)(Tz)(

*tTs

)(*

tTsM )(TzM

z

Bank of M matched filters

Matched filter output:

Observation

vector

)()( tTstrz ii

! i ,...,1!

),...,,())(),...,(),(( 2121 MM zzzTzTzTz !!z


43/200

Lecture 4 43


dttstrzi

T

i)()(

0

!

T

0

)(1 ts

T

0

ts M

Mz

z

/

1

z!

)(t

)(Tz

)(TzM

z

Bank of M correlators

Correlators output:

Observationvector

),...,,())(),...,(),(( 22 MM zzzTzTzTz !!z

Mi ,...,1!


44/200

Lecture 4 44

Today, we are going to talk about:

Detection:

Estimate the transmitted symbol based on the

received sample

Signal space used for detection Orthogonal N-dimensional space

Signal to waveform transformation and vice versa


45/200

Lecture 4 45

Signal space

What is a signal space? Vector representations of signals in an N-dimensional

orthogonal space

Why do we need a signal space?

It is a means to convert signals to vectors and vice versa.

It is a means to calculate signals energy and Euclideandistances between signals.

Why are we interested in Euclidean distances between

signals?

For detection purposes:T

he received signal is transformed toa received vectors. The signal which has the minimum

distance to the received signal is estimated as the transmitted

signal.


46/200

Lecture 4 46

Schematic example of a signal space

),()()()(

),()()()(

),()()()(

),()()()(

212211

323132321313

222122221212

121112121111

zztztztz

aatatats

aatatats

aatatats

!!

!!

!!!!

z

s

s

s

]]

]]

]]

]]

)(1 t]

)(2 t]

),( 12111 aa!s

),( 22212 aa!s

),( 32313 aa!s

),( 21 zz!z

Transmitted signal

alternatives

Received signal at

matched filter output


47/200

Lecture 4 47

Signal space

To form a signal space, first we need to knowthe inner product between two signals

(functions):

Inner (scalar) product:

Properties of inner product:

"! dtttxttx )()()(),(

= cross-correlation between x(t) and y(t)

"" )(),()(),( tytxatytax

"" )(),()(),( * tytxataytx

"""! )(),()(),()(),()( tztytztxtztytx


48/200

Lecture 4 48

Signal space

The distance in signal space is measure by calculatingthe norm.

What is norm?

Norm of a signal:

Norm between two signals:

We refer to the norm between two signals as the

Euclidean distance between two signals.

xEttxtxtxtx !!"! g

g

2

)()(),()(

)()( tata !

)()(, ttd

= length of x(t)


49/200

Lecture 4 49

Example of distances in signal space

)(1 t]

)(t] ),( 12111 aa!s

),( 22212 aa!s

),( 32313 aa!s

),( 21 zz!z

zsd ,1

zsd ,2zsd ,3

The Euclidean distance between signalsz(t) ands(t):

3,2,1

)()()()( 2222

11,

!

!!

i

zazatztd iiizsi

1E

3E

2E


50/200

Lecture 4 50

Orthogonal signal space

N-dimensional orthogonal signal space is characterized byN linearly independent functions called basisfunctions. The basis functions must satisfy the orthogonalitycondition

where

If all , the signal space is orthonormal.

_ aNjj

t1

)(!

]

jiij

T

iji Kdttttt H" )()()(),(*

0

Ttee0

Nij ,...,1, !

p

!p!

ji

jiij

0

1H

1iK


51/200

Lecture 4 51

Example of an orthonormal bases

Example: 2-dimensional orthonormal signal space

Example: 1-dimensional orthonornal signal space

1)()(

0)()()(),(

0)/2sin(2

)(

0)/2cos(2)(

21

2

0

121

2

1

!!

!"!

!

!

tt

ttttt

TtTtT

t

TtTtT

t

T

]]

]]]]

T]

T]

T t

)(1 t]

T

1

0

)(1 t]

)(t]

0

1)(1 !t] )(1 t]

0


52/200

Lecture 4 52

Signal space

Any arbitrary finite set of waveformswhere each member of the set is of duration T, can be

expressed as a linear combination of N orthonogal

waveforms where .

where

_ aMi

it

1

)(!

_ aNjj

t1

)(!

MN e

!

!N

j

jiji tats1

)()( ] Mi ,...,1!

MN e

dtttK

ttK

a

T

ji

j

ji

j

ij )()(1)(),(1 "!! ]] Ttee0Mi ,...,1! Nj ,...,1!

),...,,( 21 iNiii aaa!s2

1

ij

N

j

ji aKE !

!

Vector representation of waveform Waveform energy


53/200

Lecture 4 53

Signal space

N

j

jiji tats1

)()( ] ),...,,( 21 iNiii aaa!s

i

i

a

a

/

1

)(1 t]

)(tN]

1ia

ia

)(tsi

T

0

)(1 t]

T

0

)(tN]

iN

i

a

a

/

1

ms!)(ts

i

1i

iN

ms

Waveform to vector conversion Vector to waveform conversion

E l f j ti i l t


54/200

Lecture 4 54

Example of projecting signals to an

orthonormal signal space

),()()()(

),()()()(

),()()()(

323132321313

222122221212

121112121111

aatatats

aatatats

aatatats

!!

!!

!!

s

s

s

]]

]]

]]

)(1 t]

)(2 t]

),( 12111 aa!s

),( 22212 aa!s

),( 32313 aa!s

Transmitted signal

alternatives

dttta

T

jiij)()(

0! ] Ttee0M,...,1!Nj ,...,1!


55/200

Lecture 4 55

Signal space contd

To find an orthonormal basis functions for a given

set of signals, Gram-Schmidt procedure can beused.

Gram-Schmidt procedure: Given a signal set , compute an orthonormal basis

1. Define

2. For compute

If let

If , do not assign any basis function.

1. Renumber the basis functions such that basis is

This is only necessary if for any i in step 2.

Note that

_ aMii

ts1

)( ! _ aN

jjt

1)(

!]

)(/)(/)()( tstsEtst !!]

Mi ,...,2!

!"!

1

1

)()(),()()(i

j

jjiii tttststd ]]

0)( {ti )(/)()( ttt iii !

0)( !ti

_ a)(),...,(),( 21 ttt N]]]

0)( !tiMN e

E l f G S h id d


56/200

Lecture 4 56

Example of Gram-Schmidt procedure

Find the basis functions and plot the signal space for the following

transmitted signals:

Using Gram-Schmidt procedure:

T t

)(1 t

T t

)(2 t

)()(

)()(

)()(

21

12

11

AA

tAt

tAt

!!

!

!

ss

]

]

)(t]-A A0

1s2s

T

A

T

A

0

0

T t

)(t]

T

1

0

0)()()()(

)()()(),(

/)(/)()(

)(

122

01212

1111

0

22

11

!!

!"!

!!

!!

tAtst

Atttstts

AtsEtst

AttsE

T

T

]

]]

]

1

2


57/200

Lecture 4 57


)(tr

1)(1 tT

)( tN ]Nz

Bank of N matched filters

Observation

vector

)()( tTtrzjj

! ] Nj ,...,!

),...,,( 21 Nzzz!z!!

N

j

jiji tats1

)()( ]

MN e

Mi ,...,1!

Nz

z1

z!z


58/200

Lecture 4 58


),...,,( 21 Nzzz!z

Nj ,...,1!dtttrz jT

j )()(0

]!

T

0

)(t]

T

0

)(tN]

Nr

r

/

1

z!)(tr

1

Nz

z

f c rrel t rs

Observ ti

vect r

!!N

jjiji tat

s

1 )()( ]Mi

,...,1!

MN e

E l f t h d filt i i


59/200

Lecture 4 59

Example of matched filter receivers using

basic functions

Number of matched filters (or correlators) is reduced by 1 compared to usingmatched filters (correlators) to the transmitted signal.

T t

)(1

t

T t

)(2

t

T t

)(1

t]

T

1

0

? A1z z!)(tr z

1 matched filter

T t

)(1 t]

T

1

0

1z

T

A

T

A

0

0


60/200

Lecture 4 60

White noise in orthonormal signal space

AWGN n(t) can be expressed as)(~)()( tntntn !

Noise projected on the signal space

which impacts the detection process.

Noise outside on the signal space

"! )(),( ttnn jj ]

0)(),(~ "! tt j]

)()(1

tntnN

j

jj!

! ]

Nj ,...,1!

Nj ,...,1!

Vect r represent ti n f

),...,,( 21 Nnnn!n

)( tn

independent zero-meanGaussain random variables with

variance

_ aN

jjn 1!

2/)var( 0Nnj !


61/200

S-72.227 Digital Communication Systems

Spreadspectrum and

Code Division Multiple Access (CDMA)communications


62/200

Timo O.

orhonen,

UTCommunication Laboratory 62

Spread Spectrum Communications - Agenda Today

Basic principles and block diagrams of spread spectrum communication

systems

Characterizing concepts

Types of SS modulation: principles and circuits

direct sequence (DS)

frequency hopping (FH)

Error rates

Spreading code sequences; generation and properties

Maximal Length (a linear, cyclic code)

Gold Walsh

Asynchronous CDMA systems


63/200

Timo O.

orhonen,


How Tele-operators* Market CDMA

Coverage

ForCoverage, CDMA saveswireless carriers from deployingthe 400% more cell site thatare required by GSM

CDMAs capacity supports atleast 400% more revenue-producingsubscribers in the same spectrumwhen compared to GSM

Capacity Cost

$$A carrier who deploys CDMAinstead of GSM will havea lower capital cost

Clarity

CDMA with PureVoiceprovides wireline clarity

Choice

CDMA offers the choice of simultaneousvoice, async and packet data, FAX, andSMS.

Customer satisfaction

The Most solid foundation forattracting and retaining subscriberis based on CDMA

*From Samsumgs narrowbandCDMA (CDMAOne) marketing(2001)


64/200

Timo O.

orhonen,


Direct Sequence Spread Spectrum (DS-SS)

This figure shows BPSK-DS transmitter and receiver

(multiplication can be realized by RF-mixers)

DS-CDMA is used in WCDMA, cdma2000 and IS-95 systems

2

22

av av

A P A P ! !

spreading


65/200

Timo O.

orhonen,


Characteristics of Spread Spectrum Bandwidth of the transmitted signal Wis much greater than the original

message bandwidth (or the signaling rate R)

Transmission bandwidth is independent of the message. Applied code isknown both to the transmitter and receiver

Interference and noise immunity of SS system is larger, the larger the

processing gain

Multiple SS systems can co-exist in the same band (=CDMA). Increased

user independence (decreased interference) for(1) higher processing

gain and higher(2) code orthogonality

Spreading sequence can be very long -> enables low transmitted PSD->

low probability of interception (especially in military communications)

Narrow band signal(data)

Wideband signal(transmitted SS signal)

/ /c b c

L W R T T! !


66/200

Timo O.

orhonen,


Characteristics of Spread Spectrum (cont.)

Processing gain, in general

LargeLc

improves noise immunity, but requires a larger

transmission bandwidth

Note that DS-spread spectrum is a repetition FEC-coded systems

Jamming margin

Tells the magnitude of additional interference and noise that can be

injected to the channel without hazarding system operation.

Example:

, 10/ (1/ ) /(1/ ) / , 10 log ( )

c c b b c c dB cL W R T T T T L L! ! ! !

[ ( ) ]J c sys desp

M L L SNR!

30 dB,available processing gain2 dB,margin or system losses

10dB,required a ter despreading (at the )

18dB,additional inter erence and noise can deteriorate

receive

c

sys

desp

j

L

L

SNR

M

!!

!

!

d by this amount


67/200

Timo O.

orhonen,


Characteristics of Spread Spectrum (cont.)

Spectral efficiencyEeff: Describes how compactly TX signal fits into the

transmission band. For instance for BPSKwith some pre-filtering:

Energy efficiency (reception sensitivity): The value of

to obtain a specified error rate (often 10-9). For BPSK the error rate is

QPSK-modulation can fit twice the data rate of BPSK in the same

bandwidth. Therefore it is more energy efficient than BPSK.

/ /b T b Fff Re

R B RE B! !

,

2 2log log

1/RF fil t c cRF

b

B T LB

k M T M} } !

/ / 1/c b c c b c

L T T L T T ! !

1beff

RF b

RE

B T ! } b

T 2 2log log

c c

M M

L L!

0/

b bE NK !

2/1( 2 ), ( ) exp( 2)

2e b

k

p Q Qk dK P PT

g

! !

22 logkM k M! !

,: bandwidth for polar mod.

: number of levels

: number of bits

RF filtB

M

k


68/200

Timo O.

orhonen,


A QPSK-DS Modulator

After serial-parallel conversion (S/P) data modulates the orthogonal

carriers

Modulation on orthogonal carriers spreaded by codes c1

and c2

Spreading codes c1

and c2 may or may not be orthogonal (System

performance is independent of their orthogonality, why?)

What kind of circuit can make the demodulation (despreading)?

2 coso

P t[

2 sino

P t[

1( )c t

2( )c t/S P

( )s t( )d t

i

q

Constellationdiagram

QPSK-modulator

2 cos( ) and 2 sin( )o o

P t P t[ [


69/200

Timo O.

orhonen,


DS-CDMA (BPSK) Spectra (Tone Jamming)

Assume DS - BPSK transmission, with a single tone jamming (jamming

powerJ[W] ). The received signal is

The respective PSD of the received chip-rate signal is

At the receiverr(t) is multiplied with the local code c(t) (=despreading)

The received signal and the local code are phase-aligned:

1 0 0( ) 2 ( )cos 2 cos( ) 'd dr t Pc t T t tt J [[ U N !

_ a

2 2

0 0

0 0

1 1( ) sinc si

1 ( ) ( )2

nc2 2

c cr c cT TS f PT f f PT f f

J f f f f H H

!

1 0

0

( ) 2 ( ) cos ( )

2 cos

( )

( )

d dd

d

d t Pc t T tc t T

c t T

t

J t

[ U

[ N

!

! 1

( ) ( ) 1d d

c t T c t T

_ a

2 2

0 0

0

2 2

0 0

2 ( )cos

1 1( ) sinc s

1 1sinc sinc

inc2

2 2

2d b b b b

d

c c c c

Jc t T t

S f PT f f T PT f

JT f f T

f

J

T

T f f T

[ N

!

1 4 4 4 4 4 4 4 44 2 4 4 4 4 4 4 4 4 43

F

data

Data spectraafter phase modulator

Spreading of jammer power


70/200

Timo O.

orhonen,


Tone Jamming (cont.)

Despreading spreads the jammer power and despreads the signal power:


71/200

Timo O.

orhonen,!


Tone Jamming (cont.)

Filtering (at the BW of the phase modulator) after despreading

suppresses the jammer power:

E R t f BPSK DS S t *


72/200

Timo O."

orhonen,#


Error Rate of BPSK-DS System* DS system is a form of coding, therefore number chips, eg code weight

determines, from its own part, error rate (code gain)

Assuming that the chips are uncorrelated, prob. of code word error for abinary-block coded BPSK-DS system with code weight w is therefore

This can be expressed in terms of processing gainLcby denoting the

average signal and noise power by , respectively, yielding

Note that the symbol error rate is upper bounded due to repetition code

nature of the DS by

where tdenotes the number of erroneous bits that can be corrected in

the coded word

0

2, / ( code rate)be c m c

EP Q R w R k n

N

! ! !

0,b av b av c E P T N N T! ! ,av avP N

2 2av b av

e c m c c m

av c av

P T PP Q R w Q L R w

N T N

! !

min1

1(1 ) , ( 1)2

nm n m

esm t

n P p p t d

m

!

e !

*For further background, see J.G.Proakis:

Digital Communications (IV Ed), Section 13.2


73/200

Timo O.$

orhonen,%


Example: Error Rate of Uncoded Binary BPSK-DS

For uncoded DS w=n, thus and

We note that and yielding

Therefore, we note that increasing system processing gain W/R, error

rate can be improved

(1/ ) 1cR n n! !

0 0

2 2b b

e c m

E EP Q R w Q

N N

! !

/b av b av b E P T P R! ! 0 /av J J W !

0

/ /

/ /

b av

av av av

E P R W R

J J W J P! !

2 /

/e

av av

W RP Q

J P

!


74/200

Timo O.&

orhonen,'


Code Generation in DS-SS

DS modulator Spreading sequence period

chip interval maximal length (ML)spreading code

ML code generator

delay elements (D-flip-flops) -> XOR - circuit

- code determined by feedback taps- code rate determined by clock rate


75/200

Timo O.(

orhonen,)


Some Cyclic BlockCodes

(n,1) Repetition codes. High coding gain, but low rate

(n,k) Hamming codes. Minimum distance always 3. Thus can detect 2errors and correct one error. n=2m-1, k = n - m,

Maximum-length codes. For every integer there exists a

maximum length code (n,k) with n = 2k- 1,dmin = 2k-1.Hamming codes

are dual1 of of maximal codes.

BCH-codes. For every integer there exist a code with n = 2m-1,and where tis the error correction capability

(n,k) Reed-Solomon (RS) codes. Works with ksymbols that consist of

mbits that are encoded to yield code words ofn symbols. For these

codes and

Nowadays BCH and RS are very popular due to large dmin, large numberof codes, and easy generation

For further code references have a look on self-study material!

3ku

3umu k n mt

min2 1u d t

2 1,number of check symbols 2! !mn n k tmin

2 1! d t

1: Task: findoutfrom netwhatis meantbydualcodes!

3m u


76/200

Timo O.0

orhonen,1


Maximal Length Codes

autocorrelation

power spectral density


77/200

Timo O.2

orhonen,3


Maximal Length Codes (cont.)

Have very good autocorrelationbut cross correlation not granted

Are linear,cyclic block codes - generated by feedbacked shift registers

Number of available codes* depends on the number of shift register

stages:

Code generator design based on tables showing tap feedbacks:

5 stages->6 codes, 10 stages ->60 codes, 25 stages ->1.3x106 codes

*For the formula see: Peterson, Ziemer: Introduction to Spread Spectrum Communication, p. 121

Design of Maximal Length Generators


78/200

Timo O.4

orhonen,5


Design of Maximal Length Generators

by a Table Entry

Feedback connections can be written directly from the table:


79/200

Timo O.6

orhonen,7


Other Spreading Codes

Walsh codes: Orthogonal, used insynchronous systems, also in

WCDMA downlink Generation recursively:

All rows and columns of the matrix are orthogonal:

Gold codes: Generated by summing preferredpairs of maximal length

codes. Have a guarantee 3-level crosscorrelation:

ForN-length code there exists N+ 2 codes in a code family and

Walsh and Gold codes are used especially in multiple access systems

Gold codes are used in asynchronous communicationsbecause their

crosscorrelation is quite good as formulated above

!0 [0]H

!

1 1

11

n n

nnn

H HH

H H !

2

0 0 0 0

0 1 0 1

0 0 1 1

0 1 1 0

H

!( 1)( 1) ( 1)1 1( 1) 1 1 0

_ a ( ) / ,1/ ,( ( ) 2) /t n N N t n N

!

( 1) / 2

( 2) / 21 2 , or odd( )1 2 , or even

n

nnt nn

! 2 1 andnN


80/200

Timo O.8

orhonen,HUTCommunication Laboratory 80

Frequency Hopping Transmitter and Receiver

In FH-SS hopping frequencies are determined by the code and the

message (bits) are usually non-coherently FSK-modulated

This method is applied in BlueTooth

! dBW W

!dBW W

!sBW W

!sBW W

Frequency Hopping Spread Spectrum (FH-SS)


81/200

Timo O.9


Frequency Hopping Spread Spectrum (FH-SS)(example: transmission of two symbols/chip)

2 levelsL

2 slotsk

:chip duration

: bit duration

: symbol duration

c

b

s

T

T

T

2 ( data modulatorBW)

2 ( total FH spectral width)

L

d d

k

s d

W f

W W

! }

! }bT

2L

n p

!

4-level FSK modulation

Hopped frequencyslot determined by

hopping code


82/200

Timo O.@


Error Rate in Frequency Hopping

If there are multiple hops/symbol we have a fast-hopping system. If

there is a single hop/symbol (or below), we have a slow-hoppingsystem.

For slow-hopping non-coherent FSK-system, binary error rate is

and the respective symbol error rate is (hard-decisions)

A fast-hopping FSK system is a diversity-gain system. Assuming non-

coherent, square-law combining of respective output signals from

matched filters yields the binary error rate (withL hops/symbol)

(For further details, see J.G.Proakis: Digital Communications (IV Ed), Section 13.3)

01 exp / 2 , /2e b b bP E NK K! !

12 1 0

1

0

1exp / 2 / 2 ,

22 11

!

L i

e b i b b cL i

L i

i r

P K L

LK

ri

K K K K

!

!

! !

!

1 exp / 2 , / 12es b c cP R R k nK! !


83/200

Timo O.Korhonen,HUTCommunication Laboratory 83

DS and FH compared

FH is applicable in environments where there exist tone jammers that

can be overcame by avoiding hopping on those frequencies

DS is applicable formultiple accessbecause it allowsstatistical

multiplexing(resource reallocation) to other users (power control)

FH applies usually non-coherent modulation due to carrier

synchronization difficulties -> modulation method degrades

performance

Both methods were first used in militarycommunications,

FH can be advantageous because the hopping span can be very

large (makes eavesdroppingdifficult)

DS can be advantageous because spectral density can be much

smaller than background noise density (transmission is unnoticed)

FH is an avoidance system: does not suffer on near-fareffect!

By using hybrid systems some benefits can be combined: The system

can have a low probability of interception and negligible near-far effect

at the same time. (Differentiallycoherentmodulation is applicable)

2 710 ...10c

L p


84/200


Multiple access: FDMA, TDMA and CDMA

FDMA, TDMA and CDMA yieldconceptually the same capacity

However, in wireless communicationsCDMA has improved capacity due to

statistical multiplexing graceful degradationPerformance can still be improved by

adaptive antennas, multiuser detection,FEC, and multi-rate encoding

Example of DS multiple access waveforms


85/200


Example of DS multiple access waveforms

channel->

detecting A ... ->

polar sig.->

FDMA TDMA d CDMA d ( )


86/200


FDMA, TDMA and CDMA compared (cont.) TDMA and FDMA principle:

TDMA allocates a time instant for a user

FDMA allocates a frequency band for a user

CDMA allocates a code for user

CDMA-system can besynchronous orasynchronous:

Synchronous CDMA can not be used in multipath channels that

destroy code orthogonality Therefore, in wireless CDMA-systems as in IS-95,cdma2000,

WCDMA and IEEE 802.11 user are asynchronous

Code classification:

Orthogonal, as Walsh-codes for orthogonal or

near-orthogonal systems Near-orthogonal and non-orthogonal codes:

Gold-codes, for asynchronous systems

Maximal length codes for asynchronous systems

C i f ll l CDMA


87/200


Capacity of a cellularCDMA system

Consider uplink (MS->BS)

Each user transmitsGaussian noise (SS-signal) whose

deterministic characteristics

are stored in RX and TX

Reception and transmission

are simple multiplications Perfect power control: each

users power at the BS the same

Each user receives multiple copies of powerPrthat is other users

interference power, therefore each user receives the interference power

where Uis the number of equal power users

( 1)k rI U P! (1)

C i f ll l CDMA ( )


88/200


Each user applies a demodulator/decoder characterized by a certain

reception sensitivityEb/Io (3 - 9 dB depending on channel coding,channel, modulation method etc.)

Each user is exposed to the interference power density (assumed to be

produced by other users only)

where BTis the spreading (and RX) bandwidth

Received signal energy / bit at the signaling rate R is

Combining (1)-(3) yields the number of users

This can still be increased by using voice activity coefficient Gv = 2.67

(only about 37% of speech time effectively used), directional antennas,

for instance for a 3-way antenna GA = 2.5.

0/ [W/Hz]k TI I B! (2)

/ [ ] [ ][ ]b r

E P R J W s! ! (3)

0 0

1/ /1

1/ /

Tk o T

r b b b

R BI I B W RU

P ER E I E I ! ! ! ! (4)

Capacity of a cellularCDMA system (cont.)

( 1)k r

I P!

C it f ll l CDMA t ( t )


89/200


In cellular system neighboring cells introduce interference that decreases

capacity. It has been found out experimentally that this reduces thenumber of users by the factor

Hence asynchronous CDMA system capacity can be approximated by

yielding with the given values Gv=2.67, GA=2.4, 1+f= 1.6,

Assuming efficient error correction algorithms, dual diversity antennas,and RAKE receiver, it is possible to obtainEb/Io=6 dB = 4, and then

1 1.6f }

/

/ 1v A

b o

G GW R

E I f!

4 /

/b o

W R

E I!

WU

R} This is of order of magnitude larger value than

with the conventional (GSM;TDMA) systems!

Capacity of a cellularCDMA system (cont.)

L L d


90/200


Lessons Learned

You understand what is meant by code gain, jamming margin, and

spectral efficiency and what is their meaning in SS systems You understand how spreading and despreading works

You understand the basic principles of DS and FH systems and know

their error rates by using BPSK and FSK modulations

You know the bases of code selection for SS system. (What kind of

codes can be applied in SS systems and when they should be applied.) You understand how the capacity of asynchronous CDMA system can

be determined

March. 2007 doc.: 15-07-0624-00-004c


91/200

a c . 007 doc.: 5 07 06 00 00 c

Slide 91Submission Liang Li

Outline

One formal PHY tech proposal from Chinese

companies, universities and partners.

Major tech points are upgraded from the OQPSKModulation on IEEE 802.15.4-2006

Spreading sequence

SFD design

Pulse shaping filter

Synchronization and demodulation performance

March. 2007 doc.: 15-07-0624-00-004c


92/200


Proposed New Operation Frequency Bands

Fc= 314.3, 314.8, 315.3, 315.8 (MHz); BW=400kHz

Fc= 430.3, 430.8, 431.3, 431.8 (MHz); BW=400kHz

Fc= 433.3, 433.8, 434.3, 434.8 (MHz); BW=400kHz

Fc=780, 782,786, 788 (MHz) BW=2MHz

March. 2007 doc.: 15-07-0624-00-004c


93/200


Communication Modulation and Data Rate

PHY

(MHz)

Frequency

Bands

(MHz)

Spreading Parameters Data Parameters

Chip Rate

(Mchip/s)Modulation

Bit Rate

(kb/s)

Symbol Rate

(ksymbol/s)

315 0.4 0.2 Chirp Sequence+ MPSK 50 12.5

430 0.4 0.2Chirp Sequence

+ MPSK50 12.5

780 2 1Chirp Sequence

+ MPSK250 62.5

March. 2007 doc.: 15-07-0624-00-004c


94/200


Upgrade from PHY Layer Operation of IEEE802.15.4-2006

The new PHY tech proposal is similar to the OQPSKones used in IEEE802.15.4-2006 at sub 1GHZ on:

The principal structures of the transmitter and

receiver Operation procedure of the transceiver

The important parameters forRF parts

Some different techniques are applied in this new

PHY proposal.

March. 2007 doc.: 15-07-0624-00-004c


95/200


Reference Design of the Wireless Transceiver based on

New PHY tech Proposal

VCO

LP

witc

h

ADC

ADC

Tra eiver

LP

LNA

RF

PHY

Q

I

MAC

Bit Stream

90o

Pul e

ilter

PAI

Q

BP

Pre-

proce i gMappi g

Acqui itio

Sy c

Demodulatio

VGA

Pul e

ilter

Digital Sig al

Proce i g

Bit Stream

March. 2007 doc.: 15-07-0624-00-004c


96/200


Difference 1: Spreading Sequence and Mapping

The Direct Sequence Spread Spectrum (DSSS) tech is applied

16 orthogonal spreading sequences are designed to map 4

information bits. The base sequence is a 16 length chirp

sequence and the other 15 sequences are its cyclic shifts.

March. 2007 doc.: 15-07-0624-00-004c


97/200


Spreading Sequence

New Proposal:

Chirp code is orthogonal among its cyclic shifts

Perfect auto-correlation property of the Preamble sequence,

Perfect orthogonal property of the 16 spreading sequences,

Reduce inter-chip interference in multipath environments.

Chirp code is robust to frequency offset

Low cost implementation of transmitter and receiver.

OQPSK in 15.4-2006 in sub-1GHz :

16 sequences are quasi-orthogonal. The auto-correlation property of the Preamble sequence is notvery well.

The code is susceptible to frequency offset.

March. 2007 doc.: 15-07-0624-00-004c


98/200


Sliding Correlation Values of the Preamble Sequence

-15 -10 -5 0 5 10 15 0

2

4

6

8

10

12

14

16

Sliding Chips

Auto-correlation

Values

-15 -10 -5 0 5 10 15 0

2

4

6

8

10

12

14

16

Sliding Chips

Auto-correlation

Values

With Sequence in New Proposal With Sequence in 15.4-2006 Std

March. 2007 doc.: 15-07-0624-00-004c


99/200


Difference 2: Pre-processing on Transmission

Conjugate the first sequence to obtain SFD The SFD sequence is the conjugate of the Preamble sequence,

that means the phases of the SFD chips are adverse to thephases of the Preamble chips.

DC component removal All chips of each symbol in the head and load of PPDU should

be multiplied by 1 or 1 based on the follow PN code serials.

r6 r5 r4 r3 r2 r1 r0 PNcode

1)( 37 ! xxxG

March. 2007 doc.: 15-07-0624-00-004c


100/200

Slide100

Submission Liang Li

SFD Design

New Proposal : The 16 spreading sequences are cyclic shift of each

other.

The SFD sequence is the complex conjugate of the

first spreading sequence.

OQPSK in 15.4-2006 in sub-1GHz : The first 8 spreading sequences are cyclic shift of

each other, and the last 8 spreading sequences are

the complex conjugate of the first 8 spreadingsequences.

The SFD sequences are chosen from the spreadingsequences.

March. 2007 doc.: 15-07-0624-00-004c


101/200

Slide101

Submission Liang Li

The Pulse Shaping Filter on I and Q path is designed as a raised cosine

filter with roll-off factor 0.5:

222 /41

)/cos(

/

)/sin()(

c

c

c

c

Ttr

Ttr

Tt

Tttp

!

T

T

T

The Transmit waveform and Spectrum are :

Difference 3: Pulse Shaping on Transmission andSpectrum ofTransmit Waveform

March. 2007 doc.: 15-07-0624-00-004c


102/200

Slide102

Submission Liang Li

Pulse Shaping Filter

New Proposal: Raised cosine filterThe chip duration is 1us.The zero-to-zero bandwidth is 1.5MHz.

15.4-2

006: Half-sine filterTo O-QPSKPHY, the zero-to-zero bandwidthis 1.5MHz.To New PHY proposal, the zero-to-zerobandwidth is 3MHz.

March. 2007 doc.: 15-07-0624-00-004c


103/200

Slide103

Submission Liang Li

Synchronization Performance of

3 4 5 6 7 8 9 10 10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N

0

A

ync

Error

Rate

B -W PC

N

15.4-2006 Simulation Conditions:1) AWGN channel

environment

2) Chip rate sampling

3) Basic sliding correlation

receiver4) Synchronized on other chips

means an error

March. 2007 doc.: 15-07-0624-00-004c


104/200

Slide104

Submission Liang Li

System Packet Error Performance

3 4 5 6 7 8 9 10 10-4

10-3

10-2

10-1

100

Eb/n

0

PER

C-W PA N

15.4-2006 Simulation Conditions:1) AWGN channel

environment

2) Chip rate sampling

3) Basic sliding correlation

receiver4) Ideal synchronization5) 32 data octets in each

packet

March. 2007 doc.: 15-07-0624-00-004c


105/200

Slide105

Submission Liang Li

The Independent Simulation Results from I2R

0 2D

E F G 0 1210

-4

10-3

10-2

10-1

100

AWGNH hannel I erformance P I Q RR

Eb

/NoP d S )

T

ER

COBI16 P Ideal)

U SSS P Ideal)

COBI16 P Sync)

U SSS P Sync)

V W X Y ` a V a W

a V

b

c

a V

b 3

a V

b

d

a V

b

e

a V

f

Egh

i

p

PE

q

a YF S K ( i o r s ohere r t)

D S S S ( i o r s ohere r t)

s OB Ia Y

(i o r s ohere r t)

D S S S ( s ohere r t)

sOB I

a Y

(s

oherer

t)

I2R implements the independent Simulation based on New PHY proposal. The left figure showsits results.

BUAA obtained the Same results in the right picture. And compare with the ones of I2R on samepage. Simulation condition are

Packet Length = 20bytes; AWGN channel, Ideal Sync.

Coherent detection: Decision is based on the real parts of the correlation values

Noncoherent detection: Decision is based on the norms of the correlation values

March. 2007 doc.: 15-07-0624-00-004c


106/200

Slide106

Submission Liang Li

System Performance ofReceiver based on New Proposal

in Multipath Channel

Tapped-Delay-Line Channel Model

IEEE P802.15 Working Group for WPANs, Multipath Simulation

Models for Sub-GHz PHY Evaluation, 15-04-0585-00-004b, Oct.

2004.

Power delay profile is exponentially declined. Each path is independently Rayleigh fading.

The average power of the channel response over many packets

is 1, but in each packet the power is varied.

Short Delay Environments

Without rake receiver

Long Delay Environments

With 3-tap rake receiver

March. 2007 doc.: 15-07-0624-00-004c


107/200

Slide107

Submission Liang Li

System Performance ofRake Receiver based on New

Proposal in Multipath Channel

Test Conditions

RMS delay spread 0~600ns

Tx nonlinear amplifier, Rapps model, p=3, backoff=1.5dB

Tx and Rx frequency offset80ppm, phase noise -110dBc/Hz

@1MHz Tx and Rx IQ imbalance 2dB, 10o

3bit AD sampling, 8bit baseband processing

Rx will implement time and frequency synchronization and data

detection

5000 packets are tested for each SNR, each packet comprises20 octets

The packet error rate is counted for 90% coverage

March. 2007 doc.: 15-07-0624-00-004c


108/200

Slide108

Submission Liang Li

PER in Short Delay Environments without Rake Receiver

March. 2007 doc.: 15-07-0624-00-004c


109/200

Slide109

Submission Liang Li

PER in Long Delay Environments with 3-tap Rake Receiver

March. 2007 doc.: 15-07-0624-00-004c


110/200

Slide110

Submission Liang Li

Thank you!


111/200


112/200


113/200

1 Dr. ri M hl

INTRODUCTION


114/200

In order to transmit digital information over *bandpass channels, we have to transfer

the information to a carrier wave of.appropriate frequency

We will study some of the most commonly *

used digital modulation techniques whereinthe digital information modifies the amplitudethe phase, or the frequency of the carrier in.discrete steps

2 Dr. Uri Mahlab


115/200

OPTIMUM RECEIVERFORBINARY:DIGITAL MODULATION SCHEMS


116/200

The function of a receiver in a binary communication *system is to distinguish between two transmitted signals.S1(t) and S2(t) in the presence of noise

The performance of the receiver is usually measured *in terms of the probability of error and the receiveris said to be optimum if it yields the minimum

.probability of error

In this section, we will derive the structure of an optimum *receiver that can be used for demodulating binary

.ASK,PSK,and FSK signals

4 Dr. Uri Mahlab

Description of binary ASK,PSK, and


117/200

p y , ,: FSK schemes

-Bandpass binary data transmission system

ModulatorChannel(Hc(f

Demodulator(receiver)

{bk}

Binarydata

Input

{bk}

Transmitcarrier

Clock pulses

Noise

(n(t Clock pulses

Local carrier

Binary data output(Z(t

+

+

(V(t

+

5 Dr. Uri Mahlab

:Explanation *The input of the system is a binary bit sequence {bk} with a *

bi d bi d i T


118/200

.bit rate rb and bit duration Tb

The output of the modulator during the Kth bit interval *.depends on the Kth input bit bk

The modulator output Z(t) during the Kth bit interval is *a shifted version of one of two basic waveforms S1(t) or S2(t) and

:Z(t) is a random process defined by

bb kTtTkfor ee )1(:

!

!

! 1bif])1([

0bif])1([

)( k2

k1

b

b

Tkts

Tkts

tZ

.1

6 Dr. Uri Mahlab

The waveforms S1(t) and S2(t) have a duration *f T d h fi i h i S1( ) d S2( ) 0


119/200

of Tb and have finite energy,that is,S1(t) and S2(t) =0

],0[ bTtif and

g!

g!

b

b

T

T

dttsE

dttsE

0

2

22

0

2

11

)]([

)]([Energy:Term

7 Dr. Uri Mahlab

:The received signal + noise


120/200

dbdb

db

db

tkTttTk

tntTkt

tntTkt

tV ee

! )1(

)(])1([s

or

)(])1([s

)(

2

1

8 Dr. Uri Mahlab

Choice of signaling waveforms for various types of digital*modulation schemesS (t) S (t)=0 for

[]0[ cfT


121/200

S1(t),S2(t)=0 for

T2];,0[ c

cb fTt !

.The frequency of the carrier fc is assumed to be a multiple of rb

Type ofmodulation

ASK

PSK

FSK

bTtTS ee0);(1 bTtts ee0);(2

)sinor(cos

twAtwA

c

c

)sin(

cos

twAor

twA

c

c

0

)sin(

cos

twA

twA

c

c

}])sin{([(

})cos{(

twwAor

twwA

dc

dc

}])sin{(or[

})cos{(

twwA

twwA

dc

dc

9 Dr. Uri Mahlab

:Receiver structure


122/200

Thresholddevice or A/D

converter

(V0(t

Filter(H(f

output

Sample everyTb seconds

)()()( tntztv !

10 Dr. Uri Mahlab

:{Probability of Error-{Pe*


123/200

:{Probability of Error {Pe

The measure of performance used for comparing *!!!digital modulation schemes is the probability of error

The receiver makes errors in the decoding process *

!!! due to the noise present at its input

The receiver parameters as H(f) and threshold setting are *!!!chosen to minimize the probability of error

11 Dr. Uri Mahlab

:The output of the filter at t=kTb can be written as *


124/200

)()()( 000 bbb kTnkTskTV !

12 Dr. Uri Mahlab

:The signal component in the output at t=kTb

bkT


125/200

g

! bb dkThZkTs ]]] )()()(0

termsI I)()()1

!

]]] dkThZ b

kT

Tk

b

b

h( ) is the impulse response of the receiver filter*ISI=0*

]

!b

b

kT

Tk

bb dkThZkTs)1(

0 )()()( ]]]

13 Dr. Uri Mahlab

Substituting Z(t) from equation 1 and making*change of the variable the signal component


126/200

change of the variable, the signal component:will look like that

!!

!!!

b

b

T

bb

T

bb

b

kTsdThs

kTsdThskTs

0

k012

0k011

0

1bwhen)()()(

0bwhen)()()()(

]]]

]]]

14 Dr. Uri Mahlab

:The noise component n0(kTb) is given by *

kT


127/200

g!

bkT

bb

dkThnkTn ]]] )()()(0

he output noise n0(t) is a stationary zero mean Gaussian random process

:The variance of n0(t) is*

g

g

!! dffHfGtnEN n22

00 )()()}({

:The probability density function of n0(t) is*

gg

! nNN

nfn ;2

n-exp

2

1)(

0

2

0

0 T

15

The probability that the kth bit is incorrectly decoded*:is given by


128/200

:is given by

}1|)({21

}0|)({2

1

})(and1

)(and0{

00

00

00

00

!

!u!

!

e!!

kb

kb

bk

bke

bTkTVP

bTkTVP

TkTbor

TkTbPP.2

16 Dr. Uri Mahlab

:The conditional pdf of V0 given bk= 0 is given by*


129/200

gg

!

gg

!

!

!

0

0

2

020

0

01\

0

0

2010

0

00\

-,2

)(-exp

2

1)(

-,2

)(-exp

2

1)(

0

0

VN

s

N

Vf

VN

s

N

Vf

k

k

bV

bV

T

T

:It is similarly when bk is 1*

.3

17 Dr. Uri Mahlab

Combining equation 2 and 3 , we obtain an*i f th b bilit f P


130/200

:expression for the probability of error- Pe as

g

g

!

0

0

0

0

2

020

0

0

0

2

010

0

2

)(-exp

2

1

2

1

2

)(-exp

2

1

2

1

T

T

e

dVN

S

N

dVN

S

N

P

T

T

.4

18 Dr. Uri Mahlab

:Conditional pdf of V0 given bk


131/200

:The optimum value of the threshold T0* is*

2

0201*0

SST

!

)( 0

00 vkv bf !

)(

k0

01b

v

vf !

19Dr. Uri

Mahlab

Substituting the value of T*0 for T0 in equation 4*we can rewrite the expression for the probability


132/200

:of error as

g

g

!

!

00102

0102

2/)(

2

2/)(

0

0

2

010

0

2exp2

1

2

)(exp

2

1

Nss

ss

e

dZ

Z

dVN

sV

NP

T

T

20Dr. Uri

Mahlab

he optimum filter is the filter that maximizes*he ratio or the square of the ratio

\


133/200

qmaximizing eliminates the requirement S01


134/200

The probability of error is minimized by an *appropriate choice of h(t) which maximizes

Where

0

2

01022 )]()([

N

TsTs bb !\

!bT

bbb dThssTsTs0

120102 )()]()([)()( \\\\

And dffHfGN n

2

0 )()(g

g

!22

Dr. UriM

ahlab

If we let P(t) =S2(t)-S1(t), then the numerator of the*:quantity to be maximized is


135/200

!!!

g

g

bT

bb

bbb

dThPdThP

TPTSTS

0

00102

)()()()(

)()()(

\\\K\\

Since P(t)=0 for t


136/200

2

2

)()(

)2exp()()(

g

g

g

g!dffGfH

dffTjfPfH

n

bT

K (*)

We can maximize by applying Schwarzs*:inequality which has the form

g

g

g

g

g

g

e dffX

dffX

dffXfX2

2

2

2

1

21

)(

)(

)()((**)

2K

24Dr. Uri

Mahlab

Applying Schwarzs inequality to Equation(**) with-


137/200

)(

)2exp()()(

)()()(

2

1

fG

fTjfPfX

fGfHfX

n

b

n

T!!and

We see that H(f), which maximizes ,is given by-

)(

)2exp()(

)(

*

fG

fTjfPKfH

n

bT!

!!! Where Kis an arbitrary constant

(***)

25Dr. Uri

Mahlab

Substituting equation (***) in(*) , we obtain-:the maximum value of as 2


138/200

2K

g

g

! dffG

fP

n )(

)(2

max2K

:And the minimum probability of error is given by-

!

!

g

22exp

21 max

2

2max/

KTK

QdZZPe

26Dr. Uri

Mahlab

:Matched Filter Receiver*


139/200

If the channel noise is white, that is, Gn(f)= /2 ,then the transfer -

:function of the optimum receiver is given by

)2exp()()( * bfTjfPfH T!

From Equation (***) with the arbitrary constant K set equal to /2-:The impulse response of the optimum filter is

L

L

g

g! dfjftjfTf

Pth b )2exp()]2exp()([)(

* TT

27Dr. Uri

Mahlab

Recognizing the fact that the inverse Fourier *of P*(f) is P(-t) and that exp(-2 jfTb) representT


140/200

:a delay of Tb we obtain h(t) as T

)()( tTpth b !:Since p(t)=S1(t)-S2(t) , we have*

)()()( 12 tTStTSth bb !The impulse response h(t) is matched to the signal *:S1(t) and S2(t) and for this reason the filter is calledMATCHED FILTER

28Dr. Uri

Mahlab

:Impulse response of the Matched Filter *

(S2(t 1


141/200

(S2(t

(S1(t2 \Tb

2 \Tb

0

0

1-

2

0

Tb

t

t

t

t

t

(a)

(b)

(c)

2 \Tb(P(t)=S2(t)-S1(t

(P(-t

Tb- 02

(d)

2 \Tb0

Tb

(h(Tb-t)=p(t

2

(e)

(h(t)=p(Tb-t

29Dr. Uri

Mahlab

:Correlation Receiver*

T

The output of the receiver at t=Tb*


142/200

g !bT

bb dThVTV \\\ )()()(0

Where V( ) is the noisy input to the receiver

Substituting and noting *: that we can rewrite the preceding expression as)()()(12 \\\ ! bb TSTSh

)T(0,or0)( b! \\h

\

!

!

b b

b

T T

T

b

dSVdSV

dSSVTV

0 0

12

0

120

)()()()(

)]()()[()(

\\\\\\

\\\\(# #)

30Dr. Uri

Mahlab

Equation(# #) suggested that the optimum receiver can be implemented *as shown in Figure 1 .This form of the receiver is called

A C l ti R i


143/200

A Correlation Receiver

Thresholddevice(A\D)

integrator

integrator

-+

Sample

every Tbseconds

bT

0

bT

0

)(1 tS

)(2 tS

!

)()(

)()()(

2

1

tntS

ortntS

tV

Figure 1

31Dr. Uri

Mahlab

In actual practice, the receiver shown in Figure 1 is actually *.implemented as shown in Figure 2

In this implementation, the integrator has to be reset at the


144/200

- (end of each signaling interval in order to ovoid (I.S.I!!!Inter symbol interference

:Integrate and dump correlation receiver

Filter

tolimitnoisepower

Thresholddevice(A/D)

R(Signal z(t

+

(n(t

+

WhiteGaussian

noise

High gain

amplifier)()( 21 tStS

Closed every Tb seconds

c

Figure 2

The bandwidth of the filter preceding the integrator is assumed *!!! to be wide enough to pass z(t) without distortion

32

Example: A band pass data transmission schemeS i li h i h


145/200

uses a PSK signaling scheme with

sec2.0T,Tt0,cos)(

/10,Tt0,cos)(

bb1

b2

mtwAtS

TwtwAtS

c

bcc

!ee!

!ee! T

The carrier amplitude at the receiver input is 1 mvolt andthe psd of the A.W.G.N at input is watt/Hz. Assume

that an ideal correlation receiver is used. Calculate the.average bit error rate of the receiver

1110

33Dr. Uri

Mahlab

:Solution


146/200

34Dr. Uri

Mahlab

=Probability of error = Pe *

:Solution Continue


147/200

=Probability of error = Pe

35Dr. Uri

Mahlab

* Binary ASK signaling


148/200

Binary ASK signaling

schemes:

!

ee

!

!

1bif])1([

1)T-(k

0bif])1([

)(

k2

b

k1

b

b

b

Tkts

kTt

Tkts

tz

The binary ASK waveform can be described as

Where andtAtS

c[cos)(2 ! 0)(1 !ts

We can represent:Z(t) as

)cos)(()( tAtDtZc

[!36

Dr. UriM

ahlab

Where D(t) is a lowpass pulse waveform consisting of.rectangular pulses


149/200

.rectangular pulses

:The model for D(t) is

g

g!

!!k

bk Tktgbtd 1or0b],)1([)( k

ee

!

elswhere0

Tt01)(

btg

)()( TtdtD !

37Dr. Uri

M

ahlab

:The power spectral density is given by

2A


150/200

)()([4

)(2

c

Dc

Dz

ffGffGA

fG !

The autocorrelation function and the power spectral density:is given by

!

u

e

!

b

bD

b

bb

b

DD

Tf

fTffG

T

TT

T

R

22

2sin)(

4

1)(

or0

for44

1

)(

T

TH

\

\\

\

38Dr. Uri

M

ahlab

:The psd of Z(t) is given by


151/200

)22

22

2

2

(

)(sin

)()(sin

)()((16

)(

cb

cB

cb

cb

cz

ffT

ffT

ffTffT

ffffA

fG

!

T

T

TT

HH

39

Dr. UriM

ahlab

If we use a pulse waveform D(t) in which the individual pulsesg(t) have the shape


152/200

? A

ee

!

elsewere0

Tt0)2cos(12)(

bTT tratg

b

40Dr. Uri

M

ahlab

Coherent ASK


153/200

We start with

The signal components of the receiver output at the:of a signaling interval are

0)(andcos)( 12 !! tstAts c[

!!

!!

b

b

T

bb

T

b

TA

dttststskT

dttststskTs

0

2

122O2

0

12101

2)]()()[()(S

and

0)]()()[()(

41Dr. Uri

M

ahlab

:The optimum threshold setting in the receiver is

AkTskTs )()( 2*


154/200

b

bb TAkTskTs

T42

)()( 0201*

0

!

!

:The probability of error can be computed aseP

g

!

!

!

max2

1

22

22

max

42exp

2

1

KLT

LK

be

b

TAQdz

zp

TA

42Dr. Uri

M

ahlab

:The average signal power at the receiver input is given by

2A


155/200

4

2A

sav !We can express the probability of error in terms of the:average signal power

! L

bave TSQp

The probability of error is sometimes expressed in *: terms of the average signal energy per bit , as

bavav TsE )(!

! Lav

e

E

QP

43 Dr. Uri Mahlab

Noncoherent ASK


156/200

Noncoherent ASK:The input to the receiver is *

!

!!

0bhen)(

1bhen)(cos)(

k

k

tn

tntAtV

i

ic[

hite.andaussian,

mean,zerobetoassumedishichinputreceiverat thenoisethe)( tni

44 Dr. Uri Mahlab

Noncoharent ASKReceiver


157/200

filterbandpasstheof

outputat thenoisetheisn(t)when

0Aand1bbitdtransmitte

kthwhen theAwhere

sin)(

cos)(cos

)(cos)(

:haveoutput wefilterAt the

kk

k

!!!

!

!!

A

ttn

ttntA

tntAtY

cs

ccck

ck

[

[[

[

45

:The pdf is0r,

2exp)(

0

2

0

0| "

!!

N

r

N

rrf

kbR


158/200

0r,2

exp)(

2

0

22

0

0

0

1|

00

"

!

!N

Ar

N

ArI

N

rrf

NN

kbR

B

T

TBN

N

LL

2

filter.bandpasstheofoutputat thepo er noise

0

0

}!

!T

T

2

0

0 ))cos(exp(2

1)( duuxXI

46 Dr. Uri Mahlab

pdfs of the envelope of the noise and the signal *:pulse noise


159/200

47 Dr. Uri Mahlab

)1b|error(2

1)0b|error(

2

1kk pppe !!!

:The probability of error is given by


160/200

T

T

2

2exp

)(

ionapproximattheUsing

22

)(exp

2

1

and

8exp

2exp

where

2

1

2

1

22

2

2

00

2

0

1

2

0

2

0

2

0

0

10

x

x

xQ

N

AQdr

N

Ar

Np

N

Adr

N

r

N

rp

pp

A

e

A

e

ee

!

!

!

!

!

!

g

g

48 Dr. Uri Mahlab

1 toreducecanex,largefor p e


161/200

0

2

0

2

0

2

2

0

0

2

2

01

Aif8

exp2

1

8exp

2

41

2

1

ence,

8exp

24

NN

A

N

A

A

Np

N

AA

Np

e

e

""

}

}

}

T

T

49 Dr. Uri Mahlab

BINERY PSK SIGNALING


162/200

SCHEMES:The waveforms are *

0bforcos)(

1bforcos)(

k2

k1

!!

!!

tAts

tAts

c

c

[

[

:The binary PSK waveform Z(t) can be described by *

)cos)(()( tAtDtZc

[!.D(t) - random binary waveform *

50 Dr. Uri Mahlab

:The power spectral density of PSK signal is


163/200

b

bD

cDcDZ

Tf

fTfG

Where

ffGffGA

fG

22

2

2

sin)(

,

)]()([4

)(

T

T!

!

51 Dr. Uri Mahlab

Coherent PSK


164/200

:The signal components of the receiver output are

!!

!!

b

b

b

b

kT

Tk

bb

kT

Tk

bb

TAdttststskTs

TAdttststskTs

)1(

2

12202

)1(

2

12101

)]()()[()(

)]()()[()(

52 Dr. Uri Mahlab

:The probability of error is given by

e QP

! max

K


165/200

bav

av

av

b

e

T

bc

e

TA

E

A

E

s

TA

Qp

TAdttA

Q

b

!

!

!

!!

2

and

2s

arescheme

PSKfor thebitperenergysignal

theendpowersignalaverageThe

or

4)cos2(

2where

2

2

2

av

2

0

222

max

L

L[

LK

53 Dr. Uri Mahlab


166/200

!

!

!

L

L

av

bave

EQ

Tsp

2

2

:errorofyprobabilittheexpresscanwe

54 Dr. Uri Mahlab

DIFFERENTIALLY COHERENT *:PSK


167/200

DELAY

LOGICNETWORK

LEVELSHIFT

bT

BINERYSEQUENCE

_ a

1oro

dk

_ a1kd

1s

tAc

[cos

tAC

[coss

Z(t)

DPSK modulator

55Dr. Uri Mahlab

DPSK demodulator


168/200

Filter tolimit noisepower

Delay

Lowpassfilter orintegrator

Thresholddevice(A/D)

Z(t)

)(tn

bT

_ akb

bkTat

sample

56 Dr. Uri Mahlab

Differential encoding & decoding


169/200

In t

e e-nce

1 1 0 1 0 0 0 1 1

nco ese ence 1 1 1 0 0 1 0 1 1 1

ransmit

ase 0 0 0 i i 0 i 0 0 0

ase

om ari-son

o t t- - - -

t tit

se ence 1 1 0 1 0 0 0 1 1

57 Dr. Uri Mahlab

* BINARYFSK SIGNALING


170/200

SCHEMES ::The waveforms ofFSK signaling

1bfor)cos()(

0bfor)cos()(

k2

k1

!!

!!

ttAtS

ttAtS

dC

dc

[[

[[

:Mathematically it can be represented as

! U[[ ')'(cos)( dttDtAtZ dc

!

!!

0bfor1

1bfor1)(

k

ktD

58 Dr. Uri Mahlab

Power spectral density ofFSK

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DC Digital Communication PART7