digital communications chapeter 3. baseband demodulation/detection signal processing lab

Digital CommunicationsDigital Communications

Chapeter 3. Baseband Demodulation/Detection

Signal Processing Lab

Signal Processing Lab., http://signal.korea.ac.krDept. of Elec. and Info. Engr., Korea Univ.

Block Diagram of a DCS


Demodulation and Detection Modeling the received signal


Major Sources of Errors Inter-Symbol Interference (ISI)

Due to the filtering effect of transmitter and receiver, symbols are “smeared”.

Thermal noise (AWGN)

Disturbs the signal in an additive fashion (Additive) Has flat spectral density for all frequencies interest (White) Modeled by Gaussian random process (Gaussian Noise)


Demodulation and Detection (cont´d) Demodulation and Sampling :

Waveform recovery and preparing the received signal for detection

Improving SNR using matched filter Reducing ISI using equalizer Sampling the recovered waveform

Detection :

Estimate the transmitted symbol based on the received sample


Baseband and Bandpass

Bandpass model of detection process is equivalent to baseband model because:

The received bandpass waveform is first transformed to a baseband waveform.

Equivalence theorem:

Performing bandpass linear signal processing followed by heterodying the signal to the baseband yields the same results as heterodying the bandpass signal to the baseband followed by a baseband linear signal processing.


Likelihood


Signal Space Inner (scalar) product

Properties of inner product :

)()(

)(*)()(),(

tyandtxbetweenncorrelatiocross

dttytxtytx

)(),()(),()(),()(

)(),(*)(),(

)(),()(),(

tztytztxtztytx

tytxataytx

tytxatytax


Signal Space (cont´d)

Norm properties :

Euclidean distance between two signals :

||)(||||||)(||

)(""

|)(|)(),(||)(|| 2

txatax

txoflength

Edttxtxtxtx x

||)()(||, tytxd yx


Signal Space (cont´d) N-dimensional orthogonal signal space is characterized by N

linearly independent functions called basis functions.

The basis functions must satisfy the orthogonality condition

If all Ki=1, the signal space is orthonormal

Njj t 1)}({

Nji

Tt

ji

jiwhere

Kdttttt

ij

iji

T

jiji

,.......,1,

0

0

1

)()()(),(0

*


Signal Space (cont´d) Any arbitrary finite set of waveforms where each member of the set is of duration T, can be

expressed as a linear combination of N orthonormal waveforms where N≤M

Mmm ts 1)}({

Njj t 1)}({


Vectorial Representation


Signals and Noise


White Noise in Orthonormal Signal Space AWGN n(t) can be expressed as


Eb/No: Figure of Merit in Digital Communications SNR or S/N is the average signal power to the average noise power. SNR

should be modified in terms of bit-energy in DCS because :

Signals are transmitted within a symbol duration and hence, are energy signal (zero power)

A merit at bit-level facilitates comparison of different DCSs transmitting different number of bits per symbol.


Bit Error Probability vs Eb/No


Decision Theory


MAP and ML

)|()|( :d LikelihooMaximum

)(

)(

)|(

)|( : RatioLikelihood

)()|()()|( :Theorem Bayes'Using

)|()|( :i PosteriorA Maximum

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1

2

2

1

2211

21

1

2

1

2

1

2

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szPszP

sP

sP

szP

szP

sPszPsPszP

zsPzsP

H

H

H

H

H

H

H

H


Signal Detection Example

2

02

)(

)(

)(ln

)|(

)|(ln)(ln : Ratiolikelihood-Log

2

1)(

21

2

22

21

221

1

2

2

1

20

1

2

1

2

1

2

2

20

aaz

aaaaz

sP

sP

szp

szpz

enp

H

H

H

Hoo

H

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n

o

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Probability of Bit Error

o

o

aao

u

B

aa

az

o

aaB

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B

aaQdueP

dzedzszpP

sHPsHPP

sPsHPsPsHPP

2)(

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2

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2112

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21

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1

2

1)|(

)|()|(

)()|()()|(


Matched Filter Receiver Problem

Design the receiver filter h(t) such that the SNR (signal power to average noise power) is maximized at the sampling time.

Solution The optimum filter is the Matched filter, given by

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


Matched Filter (cont´d)

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

2max

0N

E

N

S s

T


Correlator Receiver The matched filter output at the sampling time can be realized

as the correlator output.

)(),()()()(

)]([)()(

)()()()()(

0

0

0

tstrdsrTz

dtTsrtz

dthrtrthtz

T

t

t

opt


Matched Filter and Correlator


Implementation of Matched Filter Receiver

),.......,,(

,......,1)()(

21 N

jj

rrr

NjtTtrr

r


Implementation of correlator receiver

),.......,,(

,......,1)()(

21

0

N

T

jj

rrr

Njdtttrr

r


Statistics of The Vector Signals AWGN channel model : r = si + n

Signal vector si=(si1, si2, … siN) is deterministic. Elements of noise vector n=(n1, n2, …, nN) are i, i.d Gaussian random

variables with zero-mean and variance N0/2. The noise vector pdf is

The elements of observed vector r=(r1, r2,….rN) are independent Gaussian random variables. Its pdf is

0

2

20

||||exp

)(

1)(

NNp

N

nnn

0

2

20

||||exp

)(

1)|(

NNp i

Ni

srsrn


Graphical Example of ML Detection


Average Probability of Symbol Error Erroneous decision : For the transmitted symbol mi or equivalently signal

vector si, an error in decision occurs if the observation vector r does not fall inside region Zi.

Probability of erroneous decision for a transmitted symbol

Probability of correct decision for a transmitted symbol

)| inside lienot does Pr()Pr()ˆPr( setmZsentmmm iiii r

)(1)(

)|()| inside lies Pr()(

)| inside lies Pr()Pr()ˆPr(

icie

Z iiiic

iiii

mpmp

dmpsetmZmp

setmZsentmmm

i

rrr

r

r


Avg. Prob. of Symbol Error (cont´d)

Average probability of symbol error :

For equally probable symbols :

M

iiE mmMP

1

)ˆPr()(

M

i Z

i

M

iic

M

iieE

i

dmPM

mPM

mPM

MP

1

11

)|(1

1

)(1

1)(1

)(

rrr


BER (Bit Error Rate) Received signal in Additive White Gaussian Noise Channel

After Matched Filtering & Sampling

where

)()()( tntstr i 2,1,0 iTt

2,1, inaz oi

bEa 1 bEa 2 )2/,0(: oo NNn

,


Maximum Likelihood Decision

0

2

/2

2

021

2

2exp

2

1

exp1

)|(

0 N

EQdu

u

dxN

Ex

NsHP

b

NE

o

b

o

b

duu

xQx

2exp

2

1)( where

2


BER versus Eb/No

bbobb RTWNNSTE /1,/,

WN

RS

WN

TS

N

E bbb

/

/

/0

b

b

R

W

N

S

N

E

0


Inter-Symbol Interference (ISI)


Inter-Symbol Interference (ISI) ISI in the detection process due to the filtering effects of the

system Overall equivalent system transfer function

creates echoes and hence time dispersion causes ISI at sampling time


Inter-Symbol Interference (ISI) (cont’d)

Nyquist pulses: No ISI at the sampling time Ideal Nyquist pulse:



Nyquist bandwidth constraint

Ideal Nyquist filter is not realizable. Goals and trade-off in pulse-shaping

Reduce ISI Efficient bandwidth utilization Robustness to timing error (small side lobes)

]//[222

1Hzssymbol

W

RW

R

Tss



Raised-Cosine Filter A Nyquist pulse (No ISI at the sampling time)


3.3.2 Error-Performance Degradation



Square-Root Raised Cosine (SRRC) filter and Equalizer


Types of Equalizers

Transversal filtering : Zero-forcing equalizer: Neglect the effect of noise Minimum mean square error (MSE) equalizer The basic limitation of a transversal equalizer is that it

performs poorly on channels having spectral nulls.

Decision feedback Using the past decisions to remove the ISI contributed

by them


Transversal Equalizer


Decision Feedback Equalizer

digital communications chapeter 3. baseband demodulation/detection signal processing lab

Documents

bandpass signal

energy signal

signal energy

snr signal power

average signal power

baseband model

baseband waveform

receiver filter ht